Calculus Examples

$\int \sqrt{539 + 49 x^{2}} d x$

Step 1

Let $x = \sqrt{11} \tan (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d x = \sec^{2} (t) \sqrt{11} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\sec^{2} (t) \sqrt{11}$ is positive.

$\int \sqrt{{(7 \sqrt{11})}^{2} + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{{(7 \sqrt{11})}^{2} + 49 {(\sqrt{11} \tan (t))}^{2}}$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Apply the product rule to $7 \sqrt{11}$ .

$\int \sqrt{7^{2} {\sqrt{11}}^{2} + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.2

Raise $7$ to the power of $2$ .

$\int \sqrt{49 {\sqrt{11}}^{2} + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.3

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 2.1.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\int \sqrt{49 {(11^{\frac{1}{2}})}^{2} + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.3.2

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

$\int \sqrt{49 \cdot 11^{\frac{1}{2} \cdot 2} + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.3.3

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

$\int \sqrt{49 \cdot 11^{\frac{2}{2}} + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.3.4

Cancel the common factor of $2$ .

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Step 2.1.1.3.4.1

Cancel the common factor.

$\int \sqrt{49 \cdot 11^{\frac{2}{2}} + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.3.4.2

Rewrite the expression.

$\int \sqrt{49 \cdot 11^{1} + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.3.5

Evaluate the exponent.

$\int \sqrt{49 \cdot 11 + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.4

Multiply $49$ by $11$ .

$\int \sqrt{539 + 49 {(\sqrt{11} \tan (t))}^{2}} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.5

Apply the product rule to $\sqrt{11} \tan (t)$ .

$\int \sqrt{539 + 49 ({\sqrt{11}}^{2} \tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 2.1.1.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\int \sqrt{539 + 49 ({(11^{\frac{1}{2}})}^{2} \tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.6.2

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

$\int \sqrt{539 + 49 (11^{\frac{1}{2} \cdot 2} \tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.6.3

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

$\int \sqrt{539 + 49 (11^{\frac{2}{2}} \tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.6.4

Cancel the common factor of $2$ .

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Step 2.1.1.6.4.1

Cancel the common factor.

$\int \sqrt{539 + 49 (11^{\frac{2}{2}} \tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.6.4.2

Rewrite the expression.

$\int \sqrt{539 + 49 (11^{1} \tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.6.5

Evaluate the exponent.

$\int \sqrt{539 + 49 (11 \tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.1.7

Multiply $11$ by $49$ .

$\int \sqrt{539 + 539 \tan^{2} (t)} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.2

Factor $539$ out of $539$ .

$\int \sqrt{539 (1) + 539 \tan^{2} (t)} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.3

Factor $539$ out of $539 \tan^{2} (t)$ .

$\int \sqrt{539 (1) + 539 (\tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.4

Factor $539$ out of $539 (1) + 539 (\tan^{2} (t))$ .

$\int \sqrt{539 (1 + \tan^{2} (t))} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.5

Rearrange terms.

$\int \sqrt{539 (\tan^{2} (t) + 1)} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.6

Apply pythagorean identity.

$\int \sqrt{539 \sec^{2} (t)} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.7

Rewrite $539 \sec^{2} (t)$ as ${(7 \sec (t))}^{2} \cdot 11$ .

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Step 2.1.7.1

Factor $49$ out of $539$ .

$\int \sqrt{49 (11) \sec^{2} (t)} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.7.2

Rewrite $49$ as $7^{2}$ .

$\int \sqrt{7^{2} \cdot 11 \sec^{2} (t)} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.7.3

Move $11$ .

$\int \sqrt{7^{2} \sec^{2} (t) \cdot 11} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.7.4

Rewrite $7^{2} \sec^{2} (t)$ as ${(7 \sec (t))}^{2}$ .

$\int \sqrt{{(7 \sec (t))}^{2} \cdot 11} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.1.8

Pull terms out from under the radical.

$\int 7 \sec (t) \sqrt{11} (\sec^{2} (t) \sqrt{11}) d t$

Step 2.2

Simplify.

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Step 2.2.1

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

$\int 7 (\sec^{2} (t) \sec^{1} (t)) \sqrt{11} \sqrt{11} d t$

Step 2.2.2

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

$\int 7 {\sec (t)}^{2 + 1} \sqrt{11} \sqrt{11} d t$

Step 2.2.3

Add $2$ and $1$ .

$\int 7 \sec^{3} (t) \sqrt{11} \sqrt{11} d t$

Step 2.2.4

Raise $\sqrt{11}$ to the power of $1$ .

$\int 7 \sec^{3} (t) ({\sqrt{11}}^{1} \sqrt{11}) d t$

Step 2.2.5

Raise $\sqrt{11}$ to the power of $1$ .

$\int 7 \sec^{3} (t) ({\sqrt{11}}^{1} {\sqrt{11}}^{1}) d t$

Step 2.2.6

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

$\int 7 \sec^{3} (t) {\sqrt{11}}^{1 + 1} d t$

Step 2.2.7

Add $1$ and $1$ .

$\int 7 \sec^{3} (t) {\sqrt{11}}^{2} d t$

Step 2.2.8

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 2.2.8.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\int 7 \sec^{3} (t) {(11^{\frac{1}{2}})}^{2} d t$

Step 2.2.8.2

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

$\int 7 \sec^{3} (t) \cdot 11^{\frac{1}{2} \cdot 2} d t$

Step 2.2.8.3

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

$\int 7 \sec^{3} (t) \cdot 11^{\frac{2}{2}} d t$

Step 2.2.8.4

Cancel the common factor of $2$ .

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Step 2.2.8.4.1

Cancel the common factor.

$\int 7 \sec^{3} (t) \cdot 11^{\frac{2}{2}} d t$

Step 2.2.8.4.2

Rewrite the expression.

$\int 7 \sec^{3} (t) \cdot 11^{1} d t$

Step 2.2.8.5

Evaluate the exponent.

$\int 7 \sec^{3} (t) \cdot 11 d t$

Step 2.2.9

Multiply $11$ by $7$ .

$\int 77 \sec^{3} (t) d t$

Step 3

Since $77$ is constant with respect to $t$ , move $77$ out of the integral.

$77 \int \sec^{3} (t) d t$

Step 4

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

$77 \int \sec (t) \sec^{2} (t) d t$

Step 5

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

$77 (\sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t)$

Step 6

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

$77 (\sec (t) \tan (t) - \int \tan^{1} (t) \tan (t) \sec (t) d t)$

Step 7

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

$77 (\sec (t) \tan (t) - \int \tan^{1} (t) \tan^{1} (t) \sec (t) d t)$

Step 8

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

$77 (\sec (t) \tan (t) - \int {\tan (t)}^{1 + 1} \sec (t) d t)$

Step 9

Simplify the expression.

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Step 9.1

Add $1$ and $1$ .

$77 (\sec (t) \tan (t) - \int \tan^{2} (t) \sec (t) d t)$

Step 9.2

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

$77 (\sec (t) \tan (t) - \int \sec (t) \tan^{2} (t) d t)$

Step 10

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

$77 (\sec (t) \tan (t) - \int \sec (t) (- 1 + \sec^{2} (t)) d t)$

Step 11

Simplify by multiplying through.

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Step 11.1

Rewrite the exponentiation as a product.

$77 (\sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t)$

Step 11.2

Apply the distributive property.

$77 (\sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t)$

Step 11.3

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

$77 (\sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t)$

Step 12

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

$77 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec (t) \sec (t) d t)$

Step 13

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

$77 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec^{1} (t) \sec (t) d t)$

Step 14

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

$77 (\sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{1 + 1} \sec (t) d t)$

Step 15

Add $1$ and $1$ .

$77 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec (t) d t)$

Step 16

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

$77 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec^{1} (t) d t)$

Step 17

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

$77 (\sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{2 + 1} d t)$

Step 18

Add $2$ and $1$ .

$77 (\sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 19

Split the single integral into multiple integrals.

$77 (\sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 20

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

$77 (\sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 21

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

$77 (\sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 22

Simplify by multiplying through.

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Step 22.1

Apply the distributive property.

$77 (\sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 22.2

Multiply $- 1$ by $- 1$ .

$77 (\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 23

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

$77 (\frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C)$

Step 24

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

$77 (\frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C)$

Step 25

Simplify.

$77 (\frac{1}{2}) (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 26

Combine $77$ and $\frac{1}{2}$ .

$\frac{77}{2} (\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)) + C$

Step 27

Replace all occurrences of $t$ with $\arctan (\frac{x}{\sqrt{11}})$ .

$\frac{77}{2} (\sec (\arctan (\frac{x}{\sqrt{11}})) \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28

Simplify.

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Step 28.1

Simplify each term.

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Step 28.1.1

Draw a triangle in the plane with vertices $(1, \frac{x}{\sqrt{11}})$ , $(1, 0)$ , and the origin. Then $\arctan (\frac{x}{\sqrt{11}})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(1, \frac{x}{\sqrt{11}})$ . Therefore, $\sec (\arctan (\frac{x}{\sqrt{11}}))$ is $\sqrt{1 + {(\frac{x}{\sqrt{11}})}^{2}}$ .

$\frac{77}{2} (\sqrt{1 + {(\frac{x}{\sqrt{11}})}^{2}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.2

Apply the product rule to $\frac{x}{\sqrt{11}}$ .

$\frac{77}{2} (\sqrt{1 + \frac{x^{2}}{{\sqrt{11}}^{2}}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.3

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 28.1.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{77}{2} (\sqrt{1 + \frac{x^{2}}{{(11^{\frac{1}{2}})}^{2}}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.3.2

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

$\frac{77}{2} (\sqrt{1 + \frac{x^{2}}{11^{\frac{1}{2} \cdot 2}}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.3.3

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

$\frac{77}{2} (\sqrt{1 + \frac{x^{2}}{11^{\frac{2}{2}}}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.3.4

Cancel the common factor of $2$ .

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Step 28.1.3.4.1

Cancel the common factor.

$\frac{77}{2} (\sqrt{1 + \frac{x^{2}}{11^{\frac{2}{2}}}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.3.4.2

Rewrite the expression.

$\frac{77}{2} (\sqrt{1 + \frac{x^{2}}{11^{1}}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.3.5

Evaluate the exponent.

$\frac{77}{2} (\sqrt{1 + \frac{x^{2}}{11}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.4

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

$\frac{77}{2} (\sqrt{\frac{11}{11} + \frac{x^{2}}{11}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.5

Combine the numerators over the common denominator.

$\frac{77}{2} (\sqrt{\frac{11 + x^{2}}{11}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.6

Rewrite $\sqrt{\frac{11 + x^{2}}{11}}$ as $\frac{\sqrt{11 + x^{2}}}{\sqrt{11}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}}}{\sqrt{11}} \tan (\arctan (\frac{x}{\sqrt{11}})) + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.7

The functions tangent and arctangent are inverses.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}}}{\sqrt{11}} \cdot \frac{x}{\sqrt{11}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.8

Combine.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{\sqrt{11} \sqrt{11}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.9

Simplify the denominator.

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Step 28.1.9.1

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{{\sqrt{11}}^{1} \sqrt{11}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.9.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.9.3

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{{\sqrt{11}}^{1 + 1}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.9.4

Add $1$ and $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{{\sqrt{11}}^{2}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.10

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 28.1.10.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{{(11^{\frac{1}{2}})}^{2}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.10.2

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11^{\frac{1}{2} \cdot 2}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.10.3

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11^{\frac{2}{2}}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.10.4

Cancel the common factor of $2$ .

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Step 28.1.10.4.1

Cancel the common factor.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11^{\frac{2}{2}}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.10.4.2

Rewrite the expression.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11^{1}} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.10.5

Evaluate the exponent.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sec (\arctan (\frac{x}{\sqrt{11}})) + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11

Simplify each term.

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Step 28.1.11.1

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x}{\sqrt{11}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.2

Multiply $\frac{x}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3

Combine and simplify the denominator.

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Step 28.1.11.3.1

Multiply $\frac{x}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{\sqrt{11} \sqrt{11}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.4

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{{\sqrt{11}}^{1 + 1}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.5

Add $1$ and $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{{\sqrt{11}}^{2}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 28.1.11.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.6.2

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{11^{\frac{1}{2} \cdot 2}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.6.3

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{11^{\frac{2}{2}}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.6.4

Cancel the common factor of $2$ .

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Step 28.1.11.3.6.4.1

Cancel the common factor.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{11^{\frac{2}{2}}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.6.4.2

Rewrite the expression.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{11^{1}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.3.6.5

Evaluate the exponent.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + {(\frac{x \sqrt{11}}{11})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.4

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

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Step 28.1.11.4.1

Apply the product rule to $\frac{x \sqrt{11}}{11}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{{(x \sqrt{11})}^{2}}{11^{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.4.2

Apply the product rule to $x \sqrt{11}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2} {\sqrt{11}}^{2}}{11^{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.5

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 28.1.11.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2} {(11^{\frac{1}{2}})}^{2}}{11^{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.5.2

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2} \cdot 11^{\frac{1}{2} \cdot 2}}{11^{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.5.3

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2} \cdot 11^{\frac{2}{2}}}{11^{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.5.4

Cancel the common factor of $2$ .

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Step 28.1.11.5.4.1

Cancel the common factor.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2} \cdot 11^{\frac{2}{2}}}{11^{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.5.4.2

Rewrite the expression.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2} \cdot 11^{1}}{11^{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.5.5

Evaluate the exponent.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2} \cdot 11}{11^{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.6

Raise $11$ to the power of $2$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2} \cdot 11}{121}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.7

Cancel the common factor of $11$ and $121$ .

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Step 28.1.11.7.1

Factor $11$ out of $x^{2} \cdot 11$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{11 \cdot x^{2}}{121}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.7.2

Cancel the common factors.

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Step 28.1.11.7.2.1

Factor $11$ out of $121$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{11 \cdot x^{2}}{11 \cdot 11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.7.2.2

Cancel the common factor.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{11 \cdot x^{2}}{11 \cdot 11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.7.2.3

Rewrite the expression.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{1 + \frac{x^{2}}{11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.8

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{\frac{11}{11} + \frac{x^{2}}{11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.9

Combine the numerators over the common denominator.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \sqrt{\frac{11 + x^{2}}{11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.10

Rewrite $\sqrt{\frac{11 + x^{2}}{11}}$ as $\frac{\sqrt{11 + x^{2}}}{\sqrt{11}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}}}{\sqrt{11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.11

Multiply $\frac{\sqrt{11 + x^{2}}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}}}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12

Combine and simplify the denominator.

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Step 28.1.11.12.1

Multiply $\frac{\sqrt{11 + x^{2}}}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{\sqrt{11} \sqrt{11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.4

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{{\sqrt{11}}^{1 + 1}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.5

Add $1$ and $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{{\sqrt{11}}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 28.1.11.12.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.6.2

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{11^{\frac{1}{2} \cdot 2}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.6.3

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{11^{\frac{2}{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.6.4

Cancel the common factor of $2$ .

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Step 28.1.11.12.6.4.1

Cancel the common factor.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{11^{\frac{2}{2}}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.6.4.2

Rewrite the expression.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{11^{1}} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.12.6.5

Evaluate the exponent.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 + x^{2}} \sqrt{11}}{11} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.13

Combine using the product rule for radicals.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \tan (\arctan (\frac{x}{\sqrt{11}})) |)) + C$

Step 28.1.11.14

The functions tangent and arctangent are inverses.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x}{\sqrt{11}} |)) + C$

Step 28.1.11.15

Multiply $\frac{x}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} |)) + C$

Step 28.1.11.16

Combine and simplify the denominator.

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Step 28.1.11.16.1

Multiply $\frac{x}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{\sqrt{11} \sqrt{11}} |)) + C$

Step 28.1.11.16.2

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}} |)) + C$

Step 28.1.11.16.3

Raise $\sqrt{11}$ to the power of $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}} |)) + C$

Step 28.1.11.16.4

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{{\sqrt{11}}^{1 + 1}} |)) + C$

Step 28.1.11.16.5

Add $1$ and $1$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{{\sqrt{11}}^{2}} |)) + C$

Step 28.1.11.16.6

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

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Step 28.1.11.16.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{{(11^{\frac{1}{2}})}^{2}} |)) + C$

Step 28.1.11.16.6.2

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{11^{\frac{1}{2} \cdot 2}} |)) + C$

Step 28.1.11.16.6.3

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

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{11^{\frac{2}{2}}} |)) + C$

Step 28.1.11.16.6.4

Cancel the common factor of $2$ .

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Step 28.1.11.16.6.4.1

Cancel the common factor.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{11^{\frac{2}{2}}} |)) + C$

Step 28.1.11.16.6.4.2

Rewrite the expression.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{11^{1}} |)) + C$

Step 28.1.11.16.6.5

Evaluate the exponent.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11}}{11} + \frac{x \sqrt{11}}{11} |)) + C$

Step 28.1.12

Combine the numerators over the common denominator.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{(11 + x^{2}) \cdot 11} + x \sqrt{11}}{11} |)) + C$

Step 28.1.13

Reorder factors in $\sqrt{(11 + x^{2}) \cdot 11} + x \sqrt{11}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (| \frac{\sqrt{11 (11 + x^{2})} + x \sqrt{11}}{11} |)) + C$

Step 28.1.14

Remove non-negative terms from the absolute value.

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})) + C$

Step 28.2

To write $\ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})$ as a fraction with a common denominator, multiply by $\frac{11}{11}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}) \cdot \frac{11}{11}) + C$

Step 28.3

Combine $\ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})$ and $\frac{11}{11}$ .

$\frac{77}{2} (\frac{\sqrt{11 + x^{2}} x}{11} + \frac{\ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}) \cdot 11}{11}) + C$

Step 28.4

Combine the numerators over the common denominator.

$\frac{77}{2} \cdot \frac{\sqrt{11 + x^{2}} x + \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}) \cdot 11}{11} + C$

Step 28.5

Move $11$ to the left of $\ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})$ .

$\frac{77}{2} \cdot \frac{\sqrt{11 + x^{2}} x + 11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})}{11} + C$

Step 28.6

Cancel the common factor of $11$ .

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Step 28.6.1

Factor $11$ out of $77$ .

$\frac{11 (7)}{2} \cdot \frac{\sqrt{11 + x^{2}} x + 11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})}{11} + C$

Step 28.6.2

Cancel the common factor.

$\frac{11 \cdot 7}{2} \cdot \frac{\sqrt{11 + x^{2}} x + 11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})}{11} + C$

Step 28.6.3

Rewrite the expression.

$\frac{7}{2} (\sqrt{11 + x^{2}} x + 11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})) + C$

Step 28.7

Apply the distributive property.

$\frac{7}{2} (\sqrt{11 + x^{2}} x) + \frac{7}{2} (11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})) + C$

Step 28.8

Multiply $\frac{7}{2} (\sqrt{11 + x^{2}} x)$ .

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Step 28.8.1

Combine $\sqrt{11 + x^{2}}$ and $\frac{7}{2}$ .

$\frac{\sqrt{11 + x^{2}} \cdot 7}{2} x + \frac{7}{2} (11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})) + C$

Step 28.8.2

Combine $\frac{\sqrt{11 + x^{2}} \cdot 7}{2}$ and $x$ .

$\frac{\sqrt{11 + x^{2}} \cdot 7 x}{2} + \frac{7}{2} (11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})) + C$

Step 28.9

Multiply $\frac{7}{2} (11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}))$ .

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Step 28.9.1

Combine $11$ and $\frac{7}{2}$ .

$\frac{\sqrt{11 + x^{2}} \cdot 7 x}{2} + \frac{11 \cdot 7}{2} \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}) + C$

Step 28.9.2

Multiply $11$ by $7$ .

$\frac{\sqrt{11 + x^{2}} \cdot 7 x}{2} + \frac{77}{2} \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}) + C$

Step 28.9.3

Combine $\frac{77}{2}$ and $\ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})$ .

$\frac{\sqrt{11 + x^{2}} \cdot 7 x}{2} + \frac{77 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})}{2} + C$

Step 28.10

Combine the numerators over the common denominator.

$\frac{\sqrt{11 + x^{2}} \cdot 7 x + 77 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})}{2} + C$

Step 28.11

Factor $7$ out of $\sqrt{11 + x^{2}} \cdot 7 x + 77 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})$ .

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Step 28.11.1

Factor $7$ out of $\sqrt{11 + x^{2}} \cdot 7 x$ .

$\frac{7 (\sqrt{11 + x^{2}} x) + 77 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})}{2} + C$

Step 28.11.2

Factor $7$ out of $77 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11})$ .

$\frac{7 (\sqrt{11 + x^{2}} x) + 7 (11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}))}{2} + C$

Step 28.11.3

Factor $7$ out of $7 (\sqrt{11 + x^{2}} x) + 7 (11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}))$ .

$\frac{7 (\sqrt{11 + x^{2}} x + 11 \ln (\frac{| \sqrt{11 (11 + x^{2})} + x \sqrt{11} |}{11}))}{2} + C$

Step 29

Reorder terms.

$\frac{7}{2} (\sqrt{11 + x^{2}} x + 11 \ln (\frac{1}{11} | \sqrt{11 (11 + x^{2})} + x \sqrt{11} |)) + C$