Calculus Examples

$\sqrt{5 x} \cdot \sqrt{x + 3}$

Step 1

Write $\sqrt{5 x} \cdot \sqrt{x + 3}$ as a function.

$f (x) = \sqrt{5 x} \cdot \sqrt{x + 3}$

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.

$F (x) = \int \sqrt{5 x} \cdot \sqrt{x + 3} d x$

Step 4

Combine using the product rule for radicals.

$\int \sqrt{5 x (x + 3)} d x$

Step 5

Complete the square.

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Step 5.1

Simplify the expression.

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Step 5.1.1

Apply the distributive property.

$5 x \cdot x + 5 x \cdot 3$

Step 5.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.2.1

Move $x$ .

$5 (x \cdot x) + 5 x \cdot 3$

Step 5.1.2.2

Multiply $x$ by $x$ .

$5 x^{2} + 5 x \cdot 3$

Step 5.1.3

Multiply $3$ by $5$ .

$5 x^{2} + 15 x$

Step 5.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 5$

$b = 15$

$c = 0$

Step 5.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{15}{2 \cdot 5}$

Step 5.4.2

Cancel the common factor of $15$ and $5$ .

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Step 5.4.2.1

Factor $5$ out of $15$ .

$d = \frac{5 \cdot 3}{2 \cdot 5}$

Step 5.4.2.2

Cancel the common factors.

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Step 5.4.2.2.1

Factor $5$ out of $2 \cdot 5$ .

$d = \frac{5 \cdot 3}{5 \cdot 2}$

Step 5.4.2.2.2

Cancel the common factor.

$d = \frac{5 \cdot 3}{5 \cdot 2}$

Step 5.4.2.2.3

Rewrite the expression.

$d = \frac{3}{2}$

Step 5.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{15^{2}}{4 \cdot 5}$

Step 5.5.2

Simplify the right side.

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Step 5.5.2.1

Simplify each term.

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Step 5.5.2.1.1

Raise $15$ to the power of $2$ .

$e = 0 - \frac{225}{4 \cdot 5}$

Step 5.5.2.1.2

Multiply $4$ by $5$ .

$e = 0 - \frac{225}{20}$

Step 5.5.2.1.3

Cancel the common factor of $225$ and $20$ .

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Step 5.5.2.1.3.1

Factor $5$ out of $225$ .

$e = 0 - \frac{5 (45)}{20}$

Step 5.5.2.1.3.2

Cancel the common factors.

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Step 5.5.2.1.3.2.1

Factor $5$ out of $20$ .

$e = 0 - \frac{5 \cdot 45}{5 \cdot 4}$

Step 5.5.2.1.3.2.2

Cancel the common factor.

$e = 0 - \frac{5 \cdot 45}{5 \cdot 4}$

Step 5.5.2.1.3.2.3

Rewrite the expression.

$e = 0 - \frac{45}{4}$

Step 5.5.2.2

Subtract $\frac{45}{4}$ from $0$ .

$e = - \frac{45}{4}$

Step 5.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $5 {(x + \frac{3}{2})}^{2} - \frac{45}{4}$ .

$\int \sqrt{5 {(x + \frac{3}{2})}^{2} - \frac{45}{4}} d x$

Step 6

Let $u = x + \frac{3}{2}$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = x + \frac{3}{2}$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $x + \frac{3}{2}$ .

$\frac{d}{d x} [x + \frac{3}{2}]$

Step 6.1.2

By the Sum Rule, the derivative of $x + \frac{3}{2}$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [\frac{3}{2}]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [\frac{3}{2}]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$1 + \frac{d}{d x} [\frac{3}{2}]$

Step 6.1.4

Since $\frac{3}{2}$ is constant with respect to $x$ , the derivative of $\frac{3}{2}$ with respect to $x$ is $0$ .

$1 + 0$

Step 6.1.5

Add $1$ and $0$ .

$1$

Step 6.2

Rewrite the problem using $u$ and $d u$ .

$\int \sqrt{5 u^{2} - \frac{45}{4}} d u$

Step 7

Simplify with factoring out.

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Step 7.1

Factor $5$ out of $5 u^{2} - \frac{45}{4}$ .

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Step 7.1.1

Factor $5$ out of $5 u^{2}$ .

$\int \sqrt{5 (u^{2}) - \frac{45}{4}} d u$

Step 7.1.2

Factor $5$ out of $- \frac{45}{4}$ .

$\int \sqrt{5 (u^{2}) + 5 (- \frac{9}{4})} d u$

Step 7.1.3

Factor $5$ out of $5 (u^{2}) + 5 (- \frac{9}{4})$ .

$\int \sqrt{5 (u^{2} - \frac{9}{4})} d u$

Step 7.2

Rewrite $\frac{9}{4}$ as ${(\frac{3}{2})}^{2}$ .

$\int \sqrt{5 (u^{2} - {(\frac{3}{2})}^{2})} d u$

Step 8

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = u$ and $b = \frac{3}{2}$ .

$\int \sqrt{5 (u + \frac{3}{2}) (u - \frac{3}{2})} d u$

Step 9

To write $u$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int \sqrt{5 (u \cdot \frac{2}{2} + \frac{3}{2}) (u - \frac{3}{2})} d u$

Step 10

Simplify terms.

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Step 10.1

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

$\int \sqrt{5 (\frac{u \cdot 2}{2} + \frac{3}{2}) (u - \frac{3}{2})} d u$

Step 10.2

Combine the numerators over the common denominator.

$\int \sqrt{5 \frac{u \cdot 2 + 3}{2} (u - \frac{3}{2})} d u$

Step 11

Move $2$ to the left of $u$ .

$\int \sqrt{5 \frac{2 u + 3}{2} (u - \frac{3}{2})} d u$

Step 12

To write $u$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int \sqrt{5 \frac{2 u + 3}{2} (u \cdot \frac{2}{2} - \frac{3}{2})} d u$

Step 13

Simplify terms.

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Step 13.1

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

$\int \sqrt{5 \frac{2 u + 3}{2} (\frac{u \cdot 2}{2} - \frac{3}{2})} d u$

Step 13.2

Combine the numerators over the common denominator.

$\int \sqrt{5 \frac{2 u + 3}{2} \frac{u \cdot 2 - 3}{2}} d u$

Step 14

Move $2$ to the left of $u$ .

$\int \sqrt{5 \frac{2 u + 3}{2} \frac{2 u - 3}{2}} d u$

Step 15

Combine exponents.

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Step 15.1

Combine $5$ and $\frac{2 u + 3}{2}$ .

$\int \sqrt{\frac{5 (2 u + 3)}{2} \cdot \frac{2 u - 3}{2}} d u$

Step 15.2

Multiply $\frac{5 (2 u + 3)}{2}$ by $\frac{2 u - 3}{2}$ .

$\int \sqrt{\frac{5 (2 u + 3) (2 u - 3)}{2 \cdot 2}} d u$

Step 15.3

Multiply $2$ by $2$ .

$\int \sqrt{\frac{5 (2 u + 3) (2 u - 3)}{4}} d u$

Step 16

Rewrite $\frac{5 (2 u + 3) (2 u - 3)}{4}$ as ${(\frac{1}{2})}^{2} (5 (2 u + 3) (2 u - 3))$ .

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Step 16.1

Factor the perfect power $1^{2}$ out of $5 (2 u + 3) (2 u - 3)$ .

$\int \sqrt{\frac{1^{2} (5 (2 u + 3) (2 u - 3))}{4}} d u$

Step 16.2

Factor the perfect power $2^{2}$ out of $4$ .

$\int \sqrt{\frac{1^{2} (5 (2 u + 3) (2 u - 3))}{2^{2} \cdot 1}} d u$

Step 16.3

Rearrange the fraction $\frac{1^{2} (5 (2 u + 3) (2 u - 3))}{2^{2} \cdot 1}$ .

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

Step 17

Simplify terms.

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Step 17.1

Pull terms out from under the radical.

$\int \frac{1}{2} \sqrt{5 (2 u + 3) (2 u - 3)} d u$

Step 17.2

Combine $\frac{1}{2}$ and $\sqrt{5 (2 u + 3) (2 u - 3)}$ .

$\int \frac{\sqrt{5 (2 u + 3) (2 u - 3)}}{2} d u$

Step 17.3

Apply the distributive property.

$\int \frac{\sqrt{(5 (2 u) + 5 \cdot 3) (2 u - 3)}}{2} d u$

Step 17.4

Multiply.

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Step 17.4.1

Multiply $2$ by $5$ .

$\int \frac{\sqrt{(10 u + 5 \cdot 3) (2 u - 3)}}{2} d u$

Step 17.4.2

Multiply $5$ by $3$ .

$\int \frac{\sqrt{(10 u + 15) (2 u - 3)}}{2} d u$

Step 18

Expand $(10 u + 15) (2 u - 3)$ using the FOIL Method.

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Step 18.1

Apply the distributive property.

$\int \frac{\sqrt{10 u (2 u - 3) + 15 (2 u - 3)}}{2} d u$

Step 18.2

Apply the distributive property.

$\int \frac{\sqrt{10 u (2 u) + 10 u \cdot - 3 + 15 (2 u - 3)}}{2} d u$

Step 18.3

Apply the distributive property.

$\int \frac{\sqrt{10 u (2 u) + 10 u \cdot - 3 + 15 (2 u) + 15 \cdot - 3}}{2} d u$

Step 19

Simplify and combine like terms.

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Step 19.1

Simplify each term.

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Step 19.1.1

Rewrite using the commutative property of multiplication.

$\int \frac{\sqrt{10 \cdot 2 u \cdot u + 10 u \cdot - 3 + 15 (2 u) + 15 \cdot - 3}}{2} d u$

Step 19.1.2

Multiply $u$ by $u$ by adding the exponents.

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Step 19.1.2.1

Move $u$ .

$\int \frac{\sqrt{10 \cdot 2 (u \cdot u) + 10 u \cdot - 3 + 15 (2 u) + 15 \cdot - 3}}{2} d u$

Step 19.1.2.2

Multiply $u$ by $u$ .

$\int \frac{\sqrt{10 \cdot 2 u^{2} + 10 u \cdot - 3 + 15 (2 u) + 15 \cdot - 3}}{2} d u$

Step 19.1.3

Multiply $10$ by $2$ .

$\int \frac{\sqrt{20 u^{2} + 10 u \cdot - 3 + 15 (2 u) + 15 \cdot - 3}}{2} d u$

Step 19.1.4

Multiply $- 3$ by $10$ .

$\int \frac{\sqrt{20 u^{2} - 30 u + 15 (2 u) + 15 \cdot - 3}}{2} d u$

Step 19.1.5

Multiply $2$ by $15$ .

$\int \frac{\sqrt{20 u^{2} - 30 u + 30 u + 15 \cdot - 3}}{2} d u$

Step 19.1.6

Multiply $15$ by $- 3$ .

$\int \frac{\sqrt{20 u^{2} - 30 u + 30 u - 45}}{2} d u$

Step 19.2

Add $- 30 u$ and $30 u$ .

$\int \frac{\sqrt{20 u^{2} + 0 - 45}}{2} d u$

Step 19.3

Add $20 u^{2}$ and $0$ .

$\int \frac{\sqrt{20 u^{2} - 45}}{2} d u$

Step 20

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \sqrt{20 u^{2} - 45} d u$

Step 21

Let $u = \frac{3}{2} \sec (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u = \frac{3 \sec (t) \tan (t)}{2} d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\frac{3 \sec (t) \tan (t)}{2}$ is positive.

$\frac{1}{2} \int \sqrt{{(2 \sqrt{5})}^{2} {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22

Simplify terms.

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Step 22.1

Simplify $\sqrt{{(2 \sqrt{5})}^{2} {(\frac{3}{2} \sec (t))}^{2} - 45}$ .

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Step 22.1.1

Simplify each term.

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Step 22.1.1.1

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

$\frac{1}{2} \int \sqrt{2^{2} {\sqrt{5}}^{2} {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.2

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

$\frac{1}{2} \int \sqrt{4 {\sqrt{5}}^{2} {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.3

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

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Step 22.1.1.3.1

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

$\frac{1}{2} \int \sqrt{4 {(5^{\frac{1}{2}})}^{2} {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.3.2

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

$\frac{1}{2} \int \sqrt{4 \cdot 5^{\frac{1}{2} \cdot 2} {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.3.3

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

$\frac{1}{2} \int \sqrt{4 \cdot 5^{\frac{2}{2}} {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.3.4

Cancel the common factor of $2$ .

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Step 22.1.1.3.4.1

Cancel the common factor.

$\frac{1}{2} \int \sqrt{4 \cdot 5^{\frac{2}{2}} {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.3.4.2

Rewrite the expression.

$\frac{1}{2} \int \sqrt{4 \cdot 5^{1} {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.3.5

Evaluate the exponent.

$\frac{1}{2} \int \sqrt{4 \cdot 5 {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.4

Multiply $4$ by $5$ .

$\frac{1}{2} \int \sqrt{20 {(\frac{3}{2} \sec (t))}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.5

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

$\frac{1}{2} \int \sqrt{20 {(\frac{3 \sec (t)}{2})}^{2} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.6

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

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Step 22.1.1.6.1

Apply the product rule to $\frac{3 \sec (t)}{2}$ .

$\frac{1}{2} \int \sqrt{20 \frac{{(3 \sec (t))}^{2}}{2^{2}} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.6.2

Apply the product rule to $3 \sec (t)$ .

$\frac{1}{2} \int \sqrt{20 \frac{3^{2} \sec^{2} (t)}{2^{2}} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.7

Raise $3$ to the power of $2$ .

$\frac{1}{2} \int \sqrt{20 \frac{9 \sec^{2} (t)}{2^{2}} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.8

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

$\frac{1}{2} \int \sqrt{20 \frac{9 \sec^{2} (t)}{4} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.9

Cancel the common factor of $4$ .

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Step 22.1.1.9.1

Factor $4$ out of $20$ .

$\frac{1}{2} \int \sqrt{4 (5) \frac{9 \sec^{2} (t)}{4} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.9.2

Cancel the common factor.

$\frac{1}{2} \int \sqrt{4 \cdot 5 \frac{9 \sec^{2} (t)}{4} - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.9.3

Rewrite the expression.

$\frac{1}{2} \int \sqrt{5 (9 \sec^{2} (t)) - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.1.10

Multiply $9$ by $5$ .

$\frac{1}{2} \int \sqrt{45 \sec^{2} (t) - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.2

Factor $45$ out of $45 \sec^{2} (t)$ .

$\frac{1}{2} \int \sqrt{45 (\sec^{2} (t)) - 45} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.3

Factor $45$ out of $- 45$ .

$\frac{1}{2} \int \sqrt{45 \sec^{2} (t) + 45 \cdot - 1} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.4

Factor $45$ out of $45 \sec^{2} (t) + 45 \cdot - 1$ .

$\frac{1}{2} \int \sqrt{45 (\sec^{2} (t) - 1)} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.5

Apply pythagorean identity.

$\frac{1}{2} \int \sqrt{45 \tan^{2} (t)} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.6

Rewrite $45 \tan^{2} (t)$ as ${(3 \tan (t))}^{2} \cdot 5$ .

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Step 22.1.6.1

Factor $9$ out of $45$ .

$\frac{1}{2} \int \sqrt{9 (5) \tan^{2} (t)} \frac{3 \sec (t) \tan (t)}{2} d t$

Step 22.1.6.2

Rewrite $9$ as $3^{2}$ .

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

Step 22.1.6.3

Move $5$ .

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

Step 22.1.6.4

Rewrite $3^{2} \tan^{2} (t)$ as ${(3 \tan (t))}^{2}$ .

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

Step 22.1.7

Pull terms out from under the radical.

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

Step 22.2

Simplify.

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Step 22.2.1

Combine $\frac{3 \sec (t) \tan (t)}{2}$ and $3$ .

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

Step 22.2.2

Multiply $3$ by $3$ .

$\frac{1}{2} \int \frac{9 \sec (t) \tan (t)}{2} \tan (t) \sqrt{5} d t$

Step 22.2.3

Combine $\frac{9 \sec (t) \tan (t)}{2}$ and $\tan (t)$ .

$\frac{1}{2} \int \frac{9 \sec (t) \tan (t) \tan (t)}{2} \sqrt{5} d t$

Step 22.2.4

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

$\frac{1}{2} \int \frac{9 \sec (t) (\tan^{1} (t) \tan (t))}{2} \sqrt{5} d t$

Step 22.2.5

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

$\frac{1}{2} \int \frac{9 \sec (t) (\tan^{1} (t) \tan^{1} (t))}{2} \sqrt{5} d t$

Step 22.2.6

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

$\frac{1}{2} \int \frac{9 \sec (t) {\tan (t)}^{1 + 1}}{2} \sqrt{5} d t$

Step 22.2.7

Add $1$ and $1$ .

$\frac{1}{2} \int \frac{9 \sec (t) \tan^{2} (t)}{2} \sqrt{5} d t$

Step 22.2.8

Combine $\frac{9 \sec (t) \tan^{2} (t)}{2}$ and $\sqrt{5}$ .

$\frac{1}{2} \int \frac{9 \sec (t) \tan^{2} (t) \sqrt{5}}{2} d t$

Step 23

Since $\frac{9 \sqrt{5}}{2}$ is constant with respect to $t$ , move $\frac{9 \sqrt{5}}{2}$ out of the integral.

$\frac{1}{2} (\frac{9 \sqrt{5}}{2} \int \sec (t) \tan^{2} (t) d t)$

Step 24

Simplify.

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Step 24.1

Multiply $\frac{9 \sqrt{5}}{2}$ by $\frac{1}{2}$ .

$\frac{9 \sqrt{5}}{2 \cdot 2} \int \sec (t) \tan^{2} (t) d t$

Step 24.2

Multiply $2$ by $2$ .

$\frac{9 \sqrt{5}}{4} \int \sec (t) \tan^{2} (t) d t$

Step 25

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

$\frac{9 \sqrt{5}}{4} \int \sec^{1} (t) \tan^{2} (t) d t$

Step 26

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

$\frac{9 \sqrt{5}}{4} \int \sec^{1} (t) (- 1 + \sec^{2} (t)) d t$

Step 27

Simplify terms.

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Step 27.1

Apply the distributive property.

$\frac{9 \sqrt{5}}{4} \int \sec^{1} (t) \cdot - 1 + \sec^{1} (t) \sec^{2} (t) d t$

Step 27.2

Simplify each term.

$\frac{9 \sqrt{5}}{4} \int - \sec (t) + \sec^{3} (t) d t$

Step 28

Split the single integral into multiple integrals.

$\frac{9 \sqrt{5}}{4} (\int - \sec (t) d t + \int \sec^{3} (t) d t)$

Step 29

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

$\frac{9 \sqrt{5}}{4} (- \int \sec (t) d t + \int \sec^{3} (t) d t)$

Step 30

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t)$

Step 31

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec (t) \sec^{2} (t) d t)$

Step 32

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan (t) (\sec (t) \tan (t)) d t)$

Step 33

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{1} (t) \tan (t) \sec (t) d t)$

Step 34

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{1} (t) \tan^{1} (t) \sec (t) d t)$

Step 35

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int {\tan (t)}^{1 + 1} \sec (t) d t)$

Step 36

Simplify the expression.

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Step 36.1

Add $1$ and $1$ .

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \tan^{2} (t) \sec (t) d t)$

Step 36.2

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) \tan^{2} (t) d t)$

Step 37

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec^{2} (t)) d t)$

Step 38

Simplify by multiplying through.

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Step 38.1

Rewrite the exponentiation as a product.

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) (- 1 + \sec (t) \sec (t)) d t)$

Step 38.2

Apply the distributive property.

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int \sec (t) \cdot - 1 + \sec (t) (\sec (t) \sec (t)) d t)$

Step 38.3

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \cdot \sec (t) + \sec (t) (\sec (t) \sec (t)) d t)$

Step 39

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec (t) \sec (t) d t)$

Step 40

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{1} (t) \sec^{1} (t) \sec (t) d t)$

Step 41

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{1 + 1} \sec (t) d t)$

Step 42

Add $1$ and $1$ .

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec (t) d t)$

Step 43

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{2} (t) \sec^{1} (t) d t)$

Step 44

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + {\sec (t)}^{2 + 1} d t)$

Step 45

Add $2$ and $1$ .

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - \int - 1 \sec (t) + \sec^{3} (t) d t)$

Step 46

Split the single integral into multiple integrals.

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (\int - 1 \sec (t) d t + \int \sec^{3} (t) d t))$

Step 47

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (- \int \sec (t) d t + \int \sec^{3} (t) d t))$

Step 48

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - (- (\ln (| \sec (t) + \tan (t) |) + C) + \int \sec^{3} (t) d t))$

Step 49

Simplify by multiplying through.

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Step 49.1

Apply the distributive property.

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) - - (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 49.2

Multiply $- 1$ by $- 1$ .

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C) - \int \sec^{3} (t) d t)$

Step 50

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \frac{\sec (t) \tan (t) + 1 (\ln (| \sec (t) + \tan (t) |) + C)}{2} + C)$

Step 51

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

$\frac{9 \sqrt{5}}{4} (- (\ln (| \sec (t) + \tan (t) |) + C) + \frac{\sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |) + C}{2} + C)$

Step 52

Simplify.

$\frac{9 \sqrt{5}}{4} \cdot \frac{- \ln (| \sec (t) + \tan (t) |) \cdot 2 + \sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 53

Simplify.

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Step 53.1

Multiply $2$ by $- 1$ .

$\frac{9 \sqrt{5}}{4} \cdot \frac{- 2 \ln (| \sec (t) + \tan (t) |) + \sec (t) \tan (t) + \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 53.2

Add $- 2 \ln (| \sec (t) + \tan (t) |)$ and $\ln (| \sec (t) + \tan (t) |)$ .

$\frac{9 \sqrt{5}}{4} \cdot \frac{\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)}{2} + C$

Step 53.3

Multiply $\frac{9 \sqrt{5}}{4}$ by $\frac{\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |)}{2}$ .

$\frac{9 \sqrt{5} (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))}{4 \cdot 2} + C$

Step 53.4

Multiply $4$ by $2$ .

$\frac{9 \sqrt{5} (\sec (t) \tan (t) - \ln (| \sec (t) + \tan (t) |))}{8} + C$

Step 54

Substitute back in for each integration substitution variable.

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Step 54.1

Replace all occurrences of $t$ with $arcsec (\frac{2 u}{3})$ .

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 u}{3})) \tan (arcsec (\frac{2 u}{3})) - \ln (| \sec (arcsec (\frac{2 u}{3})) + \tan (arcsec (\frac{2 u}{3})) |))}{8} + C$

Step 54.2

Replace all occurrences of $u$ with $x + \frac{3}{2}$ .

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 (x + \frac{3}{2})}{3})) \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) - \ln (| \sec (arcsec (\frac{2 (x + \frac{3}{2})}{3})) + \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) |))}{8} + C$

Step 55

Simplify.

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Step 55.1

Apply the distributive property.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 2 (\frac{3}{2})}{3})) \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) - \ln (| \sec (arcsec (\frac{2 (x + \frac{3}{2})}{3})) + \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) |))}{8} + C$

Step 55.2

Cancel the common factor of $2$ .

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Step 55.2.1

Cancel the common factor.

Step 55.2.2

Rewrite the expression.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 3}{3})) \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) - \ln (| \sec (arcsec (\frac{2 (x + \frac{3}{2})}{3})) + \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) |))}{8} + C$

Step 55.3

Apply the distributive property.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 3}{3})) \tan (arcsec (\frac{2 x + 2 (\frac{3}{2})}{3})) - \ln (| \sec (arcsec (\frac{2 (x + \frac{3}{2})}{3})) + \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) |))}{8} + C$

Step 55.4

Cancel the common factor of $2$ .

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Step 55.4.1

Cancel the common factor.

Step 55.4.2

Rewrite the expression.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 3}{3})) \tan (arcsec (\frac{2 x + 3}{3})) - \ln (| \sec (arcsec (\frac{2 (x + \frac{3}{2})}{3})) + \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) |))}{8} + C$

Step 55.5

Apply the distributive property.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 3}{3})) \tan (arcsec (\frac{2 x + 3}{3})) - \ln (| \sec (arcsec (\frac{2 x + 2 (\frac{3}{2})}{3})) + \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) |))}{8} + C$

Step 55.6

Cancel the common factor of $2$ .

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Step 55.6.1

Cancel the common factor.

Step 55.6.2

Rewrite the expression.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 3}{3})) \tan (arcsec (\frac{2 x + 3}{3})) - \ln (| \sec (arcsec (\frac{2 x + 3}{3})) + \tan (arcsec (\frac{2 (x + \frac{3}{2})}{3})) |))}{8} + C$

Step 55.7

Apply the distributive property.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 3}{3})) \tan (arcsec (\frac{2 x + 3}{3})) - \ln (| \sec (arcsec (\frac{2 x + 3}{3})) + \tan (arcsec (\frac{2 x + 2 (\frac{3}{2})}{3})) |))}{8} + C$

Step 55.8

Cancel the common factor of $2$ .

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Step 55.8.1

Cancel the common factor.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 3}{3})) \tan (arcsec (\frac{2 x + 3}{3})) - \ln (| \sec (arcsec (\frac{2 x + 3}{3})) + \tan (arcsec (\frac{2 x + 2 (\frac{3}{2})}{3})) |))}{8} + C$

Step 55.8.2

Rewrite the expression.

$\frac{9 \sqrt{5} (\sec (arcsec (\frac{2 x + 3}{3})) \tan (arcsec (\frac{2 x + 3}{3})) - \ln (| \sec (arcsec (\frac{2 x + 3}{3})) + \tan (arcsec (\frac{2 x + 3}{3})) |))}{8} + C$

Step 56

Reorder terms.

$\frac{9 \sqrt{5}}{8} (\sec (arcsec (\frac{1}{3} (2 x + 3))) \tan (arcsec (\frac{1}{3} (2 x + 3))) - \ln (| \sec (arcsec (\frac{1}{3} (2 x + 3))) + \tan (arcsec (\frac{1}{3} (2 x + 3))) |)) + C$

Step 57

The answer is the antiderivative of the function $f (x) = \sqrt{5 x} \cdot \sqrt{x + 3}$ .

$F (x) =$ $\frac{9 \sqrt{5}}{8} (\sec (arcsec (\frac{1}{3} (2 x + 3))) \tan (arcsec (\frac{1}{3} (2 x + 3))) - \ln (| \sec (arcsec (\frac{1}{3} (2 x + 3))) + \tan (arcsec (\frac{1}{3} (2 x + 3))) |)) + C$