Calculus Examples

$\int \frac{3 x^{3}}{\sqrt{x^{2} - 25}} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{x^{3}}{\sqrt{x^{2} - 25}} d x$

Step 2

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

$3 \int \frac{{(5 \sec (t))}^{3}}{\sqrt{{(5 \sec (t))}^{2} - 25}} (5 \sec (t) \tan (t)) d t$

Step 3

Simplify terms.

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Step 3.1

Simplify $\sqrt{{(5 \sec (t))}^{2} - 25}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the product rule to $5 \sec (t)$ .

$3 \int \frac{{(5 \sec (t))}^{3}}{\sqrt{5^{2} \sec^{2} (t) - 25}} (5 \sec (t) \tan (t)) d t$

Step 3.1.1.2

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

$3 \int \frac{{(5 \sec (t))}^{3}}{\sqrt{25 \sec^{2} (t) - 25}} (5 \sec (t) \tan (t)) d t$

Step 3.1.2

Factor $25$ out of $25 \sec^{2} (t)$ .

$3 \int \frac{{(5 \sec (t))}^{3}}{\sqrt{25 (\sec^{2} (t)) - 25}} (5 \sec (t) \tan (t)) d t$

Step 3.1.3

Factor $25$ out of $- 25$ .

$3 \int \frac{{(5 \sec (t))}^{3}}{\sqrt{25 \sec^{2} (t) + 25 \cdot - 1}} (5 \sec (t) \tan (t)) d t$

Step 3.1.4

Factor $25$ out of $25 \sec^{2} (t) + 25 \cdot - 1$ .

$3 \int \frac{{(5 \sec (t))}^{3}}{\sqrt{25 (\sec^{2} (t) - 1)}} (5 \sec (t) \tan (t)) d t$

Step 3.1.5

Apply pythagorean identity.

$3 \int \frac{{(5 \sec (t))}^{3}}{\sqrt{25 \tan^{2} (t)}} (5 \sec (t) \tan (t)) d t$

Step 3.1.6

Rewrite $25 \tan^{2} (t)$ as ${(5 \tan (t))}^{2}$ .

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

Step 3.1.7

Pull terms out from under the radical, assuming positive real numbers.

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

Step 3.2

Reduce the expression by cancelling the common factors.

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Step 3.2.1

Cancel the common factor of $5 \tan (t)$ .

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Step 3.2.1.1

Factor $5 \tan (t)$ out of $5 \sec (t) \tan (t)$ .

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

Step 3.2.1.2

Cancel the common factor.

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

Step 3.2.1.3

Rewrite the expression.

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

Factor $5$ out of $5 \sec (t)$ .

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

Step 3.2.2.2

Apply the product rule to $5 (\sec (t))$ .

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

Step 3.2.2.3

Raise $5$ to the power of $3$ .

$3 \int 125 \sec^{3} (t) \sec (t) d t$

Step 3.2.2.4

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

$3 \int 125 (\sec^{1} (t) \sec^{3} (t)) d t$

Step 3.2.2.5

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

$3 \int 125 {\sec (t)}^{1 + 3} d t$

Step 3.2.2.6

Add $1$ and $3$ .

$3 \int 125 \sec^{4} (t) d t$

Step 4

Since $125$ is constant with respect to $t$ , move $125$ out of the integral.

$3 (125 \int \sec^{4} (t) d t)$

Step 5

Simplify the expression.

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Step 5.1

Multiply $125$ by $3$ .

$375 \int \sec^{4} (t) d t$

Step 5.2

Rewrite $4$ as $2$ plus $2$

$375 \int {\sec (t)}^{2 + 2} d t$

Step 5.3

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

$375 \int \sec^{2} (t) \sec^{2} (t) d t$

Step 6

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

$375 \int (1 + \tan^{2} (t)) \sec^{2} (t) d t$

Step 7

Let $u = \tan (t)$ . Then $d u = \sec^{2} (t) d t$ , so $\frac{1}{\sec^{2} (t)} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1

Let $u = \tan (t)$ . Find $\frac{d u}{d t}$ .

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Step 7.1.1

Differentiate $\tan (t)$ .

$\frac{d}{d t} [\tan (t)]$

Step 7.1.2

The derivative of $\tan (t)$ with respect to $t$ is $\sec^{2} (t)$ .

$\sec^{2} (t)$

Step 7.2

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

$375 \int 1 + u^{2} d u$

Step 8

Split the single integral into multiple integrals.

$375 (\int d u + \int u^{2} d u)$

Step 9

Apply the constant rule.

$375 (u + C + \int u^{2} d u)$

Step 10

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$375 (u + C + \frac{1}{3} u^{3} + C)$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{3}$ and $u^{3}$ .

$375 (u + C + \frac{u^{3}}{3} + C)$

Step 11.2

Simplify.

$375 (u + \frac{1}{3} u^{3}) + C$

Step 12

Substitute back in for each integration substitution variable.

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Step 12.1

Replace all occurrences of $u$ with $\tan (t)$ .

$375 (\tan (t) + \frac{1}{3} \tan^{3} (t)) + C$

Step 12.2

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

$375 (\tan (arcsec (\frac{x}{5})) + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13

Simplify.

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Step 13.1

Simplify each term.

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Step 13.1.1

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

$375 (\sqrt{{(\frac{x}{5})}^{2} - 1} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.2

Rewrite $1$ as $1^{2}$ .

$375 (\sqrt{{(\frac{x}{5})}^{2} - 1^{2}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = \frac{x}{5}$ and $b = 1$ .

$375 (\sqrt{(\frac{x}{5} + 1) (\frac{x}{5} - 1)} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.4

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

$375 (\sqrt{(\frac{x}{5} + \frac{5}{5}) (\frac{x}{5} - 1)} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.5

Combine the numerators over the common denominator.

$375 (\sqrt{\frac{x + 5}{5} (\frac{x}{5} - 1)} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$375 (\sqrt{\frac{x + 5}{5} (\frac{x}{5} - 1 \cdot \frac{5}{5})} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.7

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

$375 (\sqrt{\frac{x + 5}{5} (\frac{x}{5} + \frac{- 1 \cdot 5}{5})} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.8

Combine the numerators over the common denominator.

$375 (\sqrt{\frac{x + 5}{5} \cdot \frac{x - 1 \cdot 5}{5}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.9

Multiply $- 1$ by $5$ .

$375 (\sqrt{\frac{x + 5}{5} \cdot \frac{x - 5}{5}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.10

Multiply $\frac{x + 5}{5}$ by $\frac{x - 5}{5}$ .

$375 (\sqrt{\frac{(x + 5) (x - 5)}{5 \cdot 5}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.11

Multiply $5$ by $5$ .

$375 (\sqrt{\frac{(x + 5) (x - 5)}{25}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.12

Rewrite $\frac{(x + 5) (x - 5)}{25}$ as ${(\frac{1}{5})}^{2} ((x + 5) (x - 5))$ .

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Step 13.1.12.1

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

$375 (\sqrt{\frac{1^{2} ((x + 5) (x - 5))}{25}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.12.2

Factor the perfect power $5^{2}$ out of $25$ .

$375 (\sqrt{\frac{1^{2} ((x + 5) (x - 5))}{5^{2} \cdot 1}} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.12.3

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

$375 (\sqrt{{(\frac{1}{5})}^{2} ((x + 5) (x - 5))} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.13

Pull terms out from under the radical.

$375 (\frac{1}{5} \sqrt{(x + 5) (x - 5)} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.1.14

Combine $\frac{1}{5}$ and $\sqrt{(x + 5) (x - 5)}$ .

$375 (\frac{\sqrt{(x + 5) (x - 5)}}{5} + \frac{1}{3} \tan^{3} (arcsec (\frac{x}{5}))) + C$

Step 13.2

Combine $\frac{1}{3}$ and $\tan^{3} (arcsec (\frac{x}{5}))$ .

$375 (\frac{\sqrt{(x + 5) (x - 5)}}{5} + \frac{\tan^{3} (arcsec (\frac{x}{5}))}{3}) + C$

Step 13.3

Apply the distributive property.

$375 \frac{\sqrt{(x + 5) (x - 5)}}{5} + 375 \frac{\tan^{3} (arcsec (\frac{x}{5}))}{3} + C$

Step 13.4

Cancel the common factor of $5$ .

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Step 13.4.1

Factor $5$ out of $375$ .

$5 (75) \frac{\sqrt{(x + 5) (x - 5)}}{5} + 375 \frac{\tan^{3} (arcsec (\frac{x}{5}))}{3} + C$

Step 13.4.2

Cancel the common factor.

$5 \cdot 75 \frac{\sqrt{(x + 5) (x - 5)}}{5} + 375 \frac{\tan^{3} (arcsec (\frac{x}{5}))}{3} + C$

Step 13.4.3

Rewrite the expression.

$75 \sqrt{(x + 5) (x - 5)} + 375 \frac{\tan^{3} (arcsec (\frac{x}{5}))}{3} + C$

Step 13.5

Cancel the common factor of $3$ .

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Step 13.5.1

Factor $3$ out of $375$ .

$75 \sqrt{(x + 5) (x - 5)} + 3 (125) \frac{\tan^{3} (arcsec (\frac{x}{5}))}{3} + C$

Step 13.5.2

Cancel the common factor.

$75 \sqrt{(x + 5) (x - 5)} + 3 \cdot 125 \frac{\tan^{3} (arcsec (\frac{x}{5}))}{3} + C$

Step 13.5.3

Rewrite the expression.

$75 \sqrt{(x + 5) (x - 5)} + 125 \tan^{3} (arcsec (\frac{x}{5})) + C$

Step 14

Reorder terms.

$75 \sqrt{(x + 5) (x - 5)} + 125 \tan^{3} (arcsec (\frac{1}{5} x)) + C$