Calculus Examples

$\sqrt{4 x^{2} + 5}$

Step 1

Write $\sqrt{4 x^{2} + 5}$ as a function.

$f (x) = \sqrt{4 x^{2} + 5}$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \sqrt{4 x^{2} + 5} d x$

Step 4

Let $x = \frac{\sqrt{5}}{2} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \frac{\sqrt{5} \sec^{2} (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{\sqrt{5} \sec^{2} (t)}{2}$ is positive.

$\int \sqrt{4 {(\frac{\sqrt{5}}{2} \tan (t))}^{2} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{4 {(\frac{\sqrt{5}}{2} \tan (t))}^{2} + {\sqrt{5}}^{2}}$ .

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Step 5.1.1

Simplify each term.

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Step 5.1.1.1

Combine $\frac{\sqrt{5}}{2}$ and $\tan (t)$ .

$\int \sqrt{4 {(\frac{\sqrt{5} \tan (t)}{2})}^{2} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.2.1

Apply the product rule to $\frac{\sqrt{5} \tan (t)}{2}$ .

$\int \sqrt{4 \frac{{(\sqrt{5} \tan (t))}^{2}}{2^{2}} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.2.2

Apply the product rule to $\sqrt{5} \tan (t)$ .

$\int \sqrt{4 \frac{{\sqrt{5}}^{2} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.3

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\int \sqrt{4 \frac{{(5^{\frac{1}{2}})}^{2} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.3.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \sqrt{4 \frac{5^{\frac{1}{2} \cdot 2} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.3.3

Combine $\frac{1}{2}$ and $2$ .

$\int \sqrt{4 \frac{5^{\frac{2}{2}} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.3.4

Cancel the common factor of $2$ .

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Step 5.1.1.3.4.1

Cancel the common factor.

$\int \sqrt{4 \frac{5^{\frac{2}{2}} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.3.4.2

Rewrite the expression.

$\int \sqrt{4 \frac{5^{1} \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.3.5

Evaluate the exponent.

$\int \sqrt{4 \frac{5 \tan^{2} (t)}{2^{2}} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.4

Raise $2$ to the power of $2$ .

$\int \sqrt{4 \frac{5 \tan^{2} (t)}{4} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.5

Cancel the common factor of $4$ .

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Step 5.1.1.5.1

Cancel the common factor.

$\int \sqrt{4 \frac{5 \tan^{2} (t)}{4} + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.5.2

Rewrite the expression.

$\int \sqrt{5 \tan^{2} (t) + {\sqrt{5}}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.1.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\int \sqrt{5 \tan^{2} (t) + {(5^{\frac{1}{2}})}^{2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \sqrt{5 \tan^{2} (t) + 5^{\frac{1}{2} \cdot 2}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.6.3

Combine $\frac{1}{2}$ and $2$ .

$\int \sqrt{5 \tan^{2} (t) + 5^{\frac{2}{2}}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.6.4

Cancel the common factor of $2$ .

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Step 5.1.1.6.4.1

Cancel the common factor.

$\int \sqrt{5 \tan^{2} (t) + 5^{\frac{2}{2}}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.6.4.2

Rewrite the expression.

$\int \sqrt{5 \tan^{2} (t) + 5^{1}} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.1.6.5

Evaluate the exponent.

$\int \sqrt{5 \tan^{2} (t) + 5} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.2

Factor $5$ out of $5 \tan^{2} (t) + 5$ .

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Step 5.1.2.1

Factor $5$ out of $5 \tan^{2} (t)$ .

$\int \sqrt{5 (\tan^{2} (t)) + 5} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.2.2

Factor $5$ out of $5$ .

$\int \sqrt{5 (\tan^{2} (t)) + 5 (1)} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.2.3

Factor $5$ out of $5 (\tan^{2} (t)) + 5 (1)$ .

$\int \sqrt{5 (\tan^{2} (t) + 1)} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.3

Apply pythagorean identity.

$\int \sqrt{5 \sec^{2} (t)} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.4

Reorder $5$ and $\sec^{2} (t)$ .

$\int \sqrt{\sec^{2} (t) \cdot 5} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.1.5

Pull terms out from under the radical.

$\int \sec (t) \sqrt{5} \frac{\sqrt{5} \sec^{2} (t)}{2} d t$

Step 5.2

Simplify.

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Step 5.2.1

Combine $\frac{\sqrt{5} \sec^{2} (t)}{2}$ and $\sec (t)$ .

$\int \frac{\sqrt{5} \sec^{2} (t) \sec (t)}{2} \sqrt{5} d t$

Step 5.2.2

Raise $\sec (t)$ to the power of $1$ .

$\int \frac{\sqrt{5} (\sec^{1} (t) \sec^{2} (t))}{2} \sqrt{5} d t$

Step 5.2.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{\sqrt{5} {\sec (t)}^{1 + 2}}{2} \sqrt{5} d t$

Step 5.2.4

Add $1$ and $2$ .

$\int \frac{\sqrt{5} \sec^{3} (t)}{2} \sqrt{5} d t$

Step 5.2.5

Combine $\frac{\sqrt{5} \sec^{3} (t)}{2}$ and $\sqrt{5}$ .

$\int \frac{\sqrt{5} \sec^{3} (t) \sqrt{5}}{2} d t$

Step 5.2.6

Raise $\sqrt{5}$ to the power of $1$ .

$\int \frac{{\sqrt{5}}^{1} \sqrt{5} \sec^{3} (t)}{2} d t$

Step 5.2.7

Raise $\sqrt{5}$ to the power of $1$ .

$\int \frac{{\sqrt{5}}^{1} {\sqrt{5}}^{1} \sec^{3} (t)}{2} d t$

Step 5.2.8

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{{\sqrt{5}}^{1 + 1} \sec^{3} (t)}{2} d t$

Step 5.2.9

Add $1$ and $1$ .

$\int \frac{{\sqrt{5}}^{2} \sec^{3} (t)}{2} d t$

Step 5.2.10

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 5.2.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\int \frac{{(5^{\frac{1}{2}})}^{2} \sec^{3} (t)}{2} d t$

Step 5.2.10.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int \frac{5^{\frac{1}{2} \cdot 2} \sec^{3} (t)}{2} d t$

Step 5.2.10.3

Combine $\frac{1}{2}$ and $2$ .

$\int \frac{5^{\frac{2}{2}} \sec^{3} (t)}{2} d t$

Step 5.2.10.4

Cancel the common factor of $2$ .

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Step 5.2.10.4.1

Cancel the common factor.

$\int \frac{5^{\frac{2}{2}} \sec^{3} (t)}{2} d t$

Step 5.2.10.4.2

Rewrite the expression.

$\int \frac{5^{1} \sec^{3} (t)}{2} d t$

Step 5.2.10.5

Evaluate the exponent.

$\int \frac{5 \sec^{3} (t)}{2} d t$

Step 6

Since $\frac{5}{2}$ is constant with respect to $t$ , move $\frac{5}{2}$ out of the integral.

$\frac{5}{2} \int \sec^{3} (t) d t$

Step 7

Factor $\sec (t)$ out of $\sec^{3} (t)$ .

$\frac{5}{2} \int \sec (t) \sec^{2} (t) d t$

Step 8

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \sec (t)$ and $d v = \sec^{2} (t)$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t)$

Step 9

Raise $\tan (t)$ to the power of $1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int \tan^{1} (t) \tan (t) \sec (t) d t)$

Step 10

Raise $\tan (t)$ to the power of $1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int \tan^{1} (t) \tan^{1} (t) \sec (t) d t)$

Step 11

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{2} (\sec (t) \tan (t) - \int {\tan (t)}^{1 + 1} \sec (t) d t)$

Step 12

Simplify the expression.

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Step 12.1

Add $1$ and $1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int \tan^{2} (t) \sec (t) d t)$

Step 12.2

Reorder $\tan^{2} (t)$ and $\sec (t)$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int \sec (t) \tan^{2} (t) d t)$

Step 13

Using the Pythagorean Identity, rewrite $\tan^{2} (t)$ as $- 1 + \sec^{2} (t)$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int \sec (t) (- 1 + \sec^{2} (t)) d t)$

Step 14

Simplify by multiplying through.

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Step 14.1

Rewrite the exponentiation as a product.

$\frac{5}{2} (\sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t)$

Step 14.2

Apply the distributive property.

$\frac{5}{2} (\sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t)$

Step 14.3

Reorder $\sec (t)$ and $- 1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t)$

Step 15

Raise $\sec (t)$ to the power of $1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec (t) \sec (t) d t)$

Step 16

Raise $\sec (t)$ to the power of $1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec^{1} (t) \sec (t) d t)$

Step 17

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{2} (\sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{1 + 1} \sec (t) d t)$

Step 18

Add $1$ and $1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec (t) d t)$

Step 19

Raise $\sec (t)$ to the power of $1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec^{1} (t) d t)$

Step 20

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{2} (\sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{2 + 1} d t)$

Step 21

Add $2$ and $1$ .

$\frac{5}{2} (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 22

Split the single integral into multiple integrals.

$\frac{5}{2} (\sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 23

Since $- 1$ is constant with respect to $t$ , move $- 1$ out of the integral.

$\frac{5}{2} (\sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 24

The integral of $\sec (t)$ with respect to $t$ is $\ln (| \sec (t) + \tan (t) |)$ .

$\frac{5}{2} (\sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 25

Simplify by multiplying through.

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Step 25.1

Apply the distributive property.

$\frac{5}{2} (\sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 25.2

Multiply $- 1$ by $- 1$ .

$\frac{5}{2} (\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 26

Solving for $\int \sec^{3} (t) d t$ , we find that $\int \sec^{3} (t) d t$ = $\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2}$ .

$\frac{5}{2} (\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C)$

Step 27

Multiply $\ln (| \sec (t) + \tan (t) |) + C$ by $1$ .

$\frac{5}{2} (\frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C)$

Step 28

Simplify.

$\frac{5}{2} \cdot \frac{1}{2} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 29

Simplify.

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Step 29.1

Multiply $\frac{5}{2}$ by $\frac{1}{2}$ .

$\frac{5}{2 \cdot 2} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 29.2

Multiply $2$ by $2$ .

$\frac{5}{4} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 30

Replace all occurrences of $t$ with $\arctan (\frac{2 x}{\sqrt{5}})$ .

$\frac{5}{4} (\sec (\arctan (\frac{2 x}{\sqrt{5}})) \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31

Simplify.

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Step 31.1

Simplify each term.

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Step 31.1.1

Draw a triangle in the plane with vertices $(1, \frac{2 x}{\sqrt{5}})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{2 x}{\sqrt{5}})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{2 x}{\sqrt{5}})$ . Therefore, $\sec (\arctan (\frac{2 x}{\sqrt{5}}))$ is $\sqrt{1 + {(\frac{2 x}{\sqrt{5}})}^{2}}$ .

$\frac{5}{4} (\sqrt{1 + {(\frac{2 x}{\sqrt{5}})}^{2}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 31.1.2.1

Apply the product rule to $\frac{2 x}{\sqrt{5}}$ .

$\frac{5}{4} (\sqrt{1 + \frac{{(2 x)}^{2}}{{\sqrt{5}}^{2}}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.2.2

Apply the product rule to $2 x$ .

$\frac{5}{4} (\sqrt{1 + \frac{2^{2} x^{2}}{{\sqrt{5}}^{2}}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.3

Raise $2$ to the power of $2$ .

$\frac{5}{4} (\sqrt{1 + \frac{4 x^{2}}{{\sqrt{5}}^{2}}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.4

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 31.1.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{5}{4} (\sqrt{1 + \frac{4 x^{2}}{{(5^{\frac{1}{2}})}^{2}}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.4.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{4} (\sqrt{1 + \frac{4 x^{2}}{5^{\frac{1}{2} \cdot 2}}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.4.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{4} (\sqrt{1 + \frac{4 x^{2}}{5^{\frac{2}{2}}}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.4.4

Cancel the common factor of $2$ .

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Step 31.1.4.4.1

Cancel the common factor.

$\frac{5}{4} (\sqrt{1 + \frac{4 x^{2}}{5^{\frac{2}{2}}}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.4.4.2

Rewrite the expression.

$\frac{5}{4} (\sqrt{1 + \frac{4 x^{2}}{5^{1}}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.4.5

Evaluate the exponent.

$\frac{5}{4} (\sqrt{1 + \frac{4 x^{2}}{5}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.5

Write $1$ as a fraction with a common denominator.

$\frac{5}{4} (\sqrt{\frac{5}{5} + \frac{4 x^{2}}{5}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.6

Combine the numerators over the common denominator.

$\frac{5}{4} (\sqrt{\frac{5 + 4 x^{2}}{5}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.7

Rewrite $\sqrt{\frac{5 + 4 x^{2}}{5}}$ as $\frac{\sqrt{5 + 4 x^{2}}}{\sqrt{5}}$ .

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}}}{\sqrt{5}} \tan (\arctan (\frac{2 x}{\sqrt{5}})) + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.8

The functions tangent and arctangent are inverses.

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}}}{\sqrt{5}} \cdot \frac{2 x}{\sqrt{5}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.9

Combine.

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} (2 x)}{\sqrt{5} \sqrt{5}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.10

Simplify the denominator.

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Step 31.1.10.1

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{{\sqrt{5}}^{1} \sqrt{5}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.10.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.10.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{{\sqrt{5}}^{1 + 1}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.10.4

Add $1$ and $1$ .

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{{\sqrt{5}}^{2}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.11

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 31.1.11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{{(5^{\frac{1}{2}})}^{2}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.11.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{5^{\frac{1}{2} \cdot 2}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.11.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{5^{\frac{2}{2}}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.11.4

Cancel the common factor of $2$ .

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Step 31.1.11.4.1

Cancel the common factor.

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{5^{\frac{2}{2}}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.11.4.2

Rewrite the expression.

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{5^{1}} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.11.5

Evaluate the exponent.

$\frac{5}{4} (\frac{\sqrt{5 + 4 x^{2}} \cdot 2 x}{5} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.12

Move $2$ to the left of $\sqrt{5 + 4 x^{2}}$ .

$\frac{5}{4} (\frac{2 \cdot \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sec (\arctan (\frac{2 x}{\sqrt{5}})) + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13

Simplify each term.

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Step 31.1.13.1

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x}{\sqrt{5}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.2

Multiply $\frac{2 x}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3

Combine and simplify the denominator.

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Step 31.1.13.3.1

Multiply $\frac{2 x}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{\sqrt{5} \sqrt{5}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{{\sqrt{5}}^{1 + 1}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.5

Add $1$ and $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{{\sqrt{5}}^{2}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 31.1.13.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{5^{\frac{1}{2} \cdot 2}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{5^{\frac{2}{2}}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.6.4

Cancel the common factor of $2$ .

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Step 31.1.13.3.6.4.1

Cancel the common factor.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{5^{\frac{2}{2}}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.6.4.2

Rewrite the expression.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{5^{1}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.3.6.5

Evaluate the exponent.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + {(\frac{2 x \sqrt{5}}{5})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 31.1.13.4.1

Apply the product rule to $\frac{2 x \sqrt{5}}{5}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{{(2 x \sqrt{5})}^{2}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.4.2

Apply the product rule to $2 x \sqrt{5}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{{(2 x)}^{2} {\sqrt{5}}^{2}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.4.3

Apply the product rule to $2 x$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{2^{2} x^{2} {\sqrt{5}}^{2}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.5

Simplify the numerator.

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Step 31.1.13.5.1

Raise $2$ to the power of $2$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{4 x^{2} {\sqrt{5}}^{2}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.5.2

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 31.1.13.5.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{4 x^{2} {(5^{\frac{1}{2}})}^{2}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.5.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{4 x^{2} \cdot 5^{\frac{1}{2} \cdot 2}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.5.2.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{4 x^{2} \cdot 5^{\frac{2}{2}}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.5.2.4

Cancel the common factor of $2$ .

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Step 31.1.13.5.2.4.1

Cancel the common factor.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{4 x^{2} \cdot 5^{\frac{2}{2}}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.5.2.4.2

Rewrite the expression.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{4 x^{2} \cdot 5^{1}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.5.2.5

Evaluate the exponent.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{4 x^{2} \cdot 5}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.5.3

Multiply $5$ by $4$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{20 x^{2}}{5^{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.6

Raise $5$ to the power of $2$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{20 x^{2}}{25}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.7

Cancel the common factor of $20$ and $25$ .

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Step 31.1.13.7.1

Factor $5$ out of $20 x^{2}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{5 (4 x^{2})}{25}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.7.2

Cancel the common factors.

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Step 31.1.13.7.2.1

Factor $5$ out of $25$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{5 (4 x^{2})}{5 (5)}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.7.2.2

Cancel the common factor.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{5 (4 x^{2})}{5 \cdot 5}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.7.2.3

Rewrite the expression.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{1 + \frac{4 x^{2}}{5}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.8

Write $1$ as a fraction with a common denominator.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{\frac{5}{5} + \frac{4 x^{2}}{5}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.9

Combine the numerators over the common denominator.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \sqrt{\frac{5 + 4 x^{2}}{5}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.10

Rewrite $\sqrt{\frac{5 + 4 x^{2}}{5}}$ as $\frac{\sqrt{5 + 4 x^{2}}}{\sqrt{5}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}}}{\sqrt{5}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.11

Multiply $\frac{\sqrt{5 + 4 x^{2}}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}}}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12

Combine and simplify the denominator.

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Step 31.1.13.12.1

Multiply $\frac{\sqrt{5 + 4 x^{2}}}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{\sqrt{5} \sqrt{5}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{{\sqrt{5}}^{1 + 1}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.5

Add $1$ and $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{{\sqrt{5}}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 31.1.13.12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{5^{\frac{2}{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.6.4

Cancel the common factor of $2$ .

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Step 31.1.13.12.6.4.1

Cancel the common factor.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{5^{\frac{2}{2}}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.6.4.2

Rewrite the expression.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{5^{1}} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.12.6.5

Evaluate the exponent.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 + 4 x^{2}} \sqrt{5}}{5} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.13

Combine using the product rule for radicals.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \tan (\arctan (\frac{2 x}{\sqrt{5}})) |)) + C$

Step 31.1.13.14

The functions tangent and arctangent are inverses.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x}{\sqrt{5}} |)) + C$

Step 31.1.13.15

Multiply $\frac{2 x}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}} |)) + C$

Step 31.1.13.16

Combine and simplify the denominator.

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Step 31.1.13.16.1

Multiply $\frac{2 x}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{\sqrt{5} \sqrt{5}} |)) + C$

Step 31.1.13.16.2

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}} |)) + C$

Step 31.1.13.16.3

Raise $\sqrt{5}$ to the power of $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}} |)) + C$

Step 31.1.13.16.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{{\sqrt{5}}^{1 + 1}} |)) + C$

Step 31.1.13.16.5

Add $1$ and $1$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{{\sqrt{5}}^{2}} |)) + C$

Step 31.1.13.16.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 31.1.13.16.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{{(5^{\frac{1}{2}})}^{2}} |)) + C$

Step 31.1.13.16.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{5^{\frac{1}{2} \cdot 2}} |)) + C$

Step 31.1.13.16.6.3

Combine $\frac{1}{2}$ and $2$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{5^{\frac{2}{2}}} |)) + C$

Step 31.1.13.16.6.4

Cancel the common factor of $2$ .

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Step 31.1.13.16.6.4.1

Cancel the common factor.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{5^{\frac{2}{2}}} |)) + C$

Step 31.1.13.16.6.4.2

Rewrite the expression.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{5^{1}} |)) + C$

Step 31.1.13.16.6.5

Evaluate the exponent.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5}}{5} + \frac{2 x \sqrt{5}}{5} |)) + C$

Step 31.1.14

Combine the numerators over the common denominator.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{(5 + 4 x^{2}) \cdot 5} + 2 x \sqrt{5}}{5} |)) + C$

Step 31.1.15

Reorder factors in $\sqrt{(5 + 4 x^{2}) \cdot 5} + 2 x \sqrt{5}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (| \frac{\sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5}}{5} |)) + C$

Step 31.1.16

Remove non-negative terms from the absolute value.

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.2

To write $\ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5}) \cdot \frac{5}{5}) + C$

Step 31.3

Combine $\ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})$ and $\frac{5}{5}$ .

$\frac{5}{4} (\frac{2 \sqrt{5 + 4 x^{2}} x}{5} + \frac{\ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5}) \cdot 5}{5}) + C$

Step 31.4

Combine the numerators over the common denominator.

$\frac{5}{4} \cdot \frac{2 \sqrt{5 + 4 x^{2}} x + \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5}) \cdot 5}{5} + C$

Step 31.5

Move $5$ to the left of $\ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})$ .

$\frac{5}{4} \cdot \frac{2 \sqrt{5 + 4 x^{2}} x + 5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})}{5} + C$

Step 31.6

Cancel the common factor of $5$ .

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Step 31.6.1

Cancel the common factor.

$\frac{5}{4} \cdot \frac{2 \sqrt{5 + 4 x^{2}} x + 5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})}{5} + C$

Step 31.6.2

Rewrite the expression.

$\frac{1}{4} (2 \sqrt{5 + 4 x^{2}} x + 5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.7

Apply the distributive property.

$\frac{1}{4} (2 \sqrt{5 + 4 x^{2}} x) + \frac{1}{4} (5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.8

Cancel the common factor of $2$ .

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Step 31.8.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (2 \sqrt{5 + 4 x^{2}} x) + \frac{1}{4} (5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.8.2

Factor $2$ out of $2 \sqrt{5 + 4 x^{2}} x$ .

$\frac{1}{2 (2)} (2 (\sqrt{5 + 4 x^{2}} x)) + \frac{1}{4} (5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.8.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 (\sqrt{5 + 4 x^{2}} x)) + \frac{1}{4} (5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.8.4

Rewrite the expression.

$\frac{1}{2} (\sqrt{5 + 4 x^{2}} x) + \frac{1}{4} (5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.9

Combine $\sqrt{5 + 4 x^{2}}$ and $\frac{1}{2}$ .

$\frac{\sqrt{5 + 4 x^{2}}}{2} x + \frac{1}{4} (5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.10

Combine $\frac{\sqrt{5 + 4 x^{2}}}{2}$ and $x$ .

$\frac{\sqrt{5 + 4 x^{2}} x}{2} + \frac{1}{4} (5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})) + C$

Step 31.11

Multiply $\frac{1}{4} (5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5}))$ .

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Step 31.11.1

Combine $5$ and $\frac{1}{4}$ .

$\frac{\sqrt{5 + 4 x^{2}} x}{2} + \frac{5}{4} \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5}) + C$

Step 31.11.2

Combine $\frac{5}{4}$ and $\ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})$ .

$\frac{\sqrt{5 + 4 x^{2}} x}{2} + \frac{5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})}{4} + C$

Step 31.12

To write $\frac{\sqrt{5 + 4 x^{2}} x}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\sqrt{5 + 4 x^{2}} x}{2} \cdot \frac{2}{2} + \frac{5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})}{4} + C$

Step 31.13

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 31.13.1

Multiply $\frac{\sqrt{5 + 4 x^{2}} x}{2}$ by $\frac{2}{2}$ .

$\frac{\sqrt{5 + 4 x^{2}} x \cdot 2}{2 \cdot 2} + \frac{5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})}{4} + C$

Step 31.13.2

Multiply $2$ by $2$ .

$\frac{\sqrt{5 + 4 x^{2}} x \cdot 2}{4} + \frac{5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})}{4} + C$

Step 31.14

Combine the numerators over the common denominator.

$\frac{\sqrt{5 + 4 x^{2}} x \cdot 2 + 5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})}{4} + C$

Step 31.15

Move $2$ to the left of $\sqrt{5 + 4 x^{2}} x$ .

$\frac{2 \sqrt{5 + 4 x^{2}} x + 5 \ln (\frac{| \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |}{5})}{4} + C$

Step 32

Reorder terms.

$\frac{1}{4} (2 \sqrt{5 + 4 x^{2}} x + 5 \ln (\frac{1}{5} | \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |)) + C$

Step 33

The answer is the antiderivative of the function $f (x) = \sqrt{4 x^{2} + 5}$ .

$F (x) =$ $\frac{1}{4} (2 \sqrt{5 + 4 x^{2}} x + 5 \ln (\frac{1}{5} | \sqrt{5 (5 + 4 x^{2})} + 2 x \sqrt{5} |)) + C$