Calculus Examples

$\int x^{3} \sqrt{x^{2} - 1} d x$

Step 1

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

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

Step 2

Simplify terms.

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Step 2.1

Simplify $\sqrt{\sec^{2} (t) - 1}$ .

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Step 2.1.1

Apply pythagorean identity.

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

Step 2.1.2

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

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

Step 2.2

Simplify the expression.

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Step 2.2.1

Simplify.

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Step 2.2.1.1

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

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

Step 2.2.1.2

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

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

Step 2.2.1.3

Add $1$ and $3$ .

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

Step 2.2.1.4

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

$\int \sec^{4} (t) (\tan^{1} (t) \tan (t)) d t$

Step 2.2.1.5

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

$\int \sec^{4} (t) (\tan^{1} (t) \tan^{1} (t)) d t$

Step 2.2.1.6

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

$\int \sec^{4} (t) {\tan (t)}^{1 + 1} d t$

Step 2.2.1.7

Add $1$ and $1$ .

$\int \sec^{4} (t) \tan^{2} (t) d t$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

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

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

Step 2.2.2.2

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

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

Step 3

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

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

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $\tan (t)$ .

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

Step 4.1.2

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

$\sec^{2} (t)$

Step 4.2

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

$\int (1 + u^{2}) u^{2} d u$

Step 5

Multiply $(1 + u^{2}) u^{2}$ .

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

Step 6

Simplify.

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Step 6.1

Multiply $u^{2}$ by $1$ .

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

Step 6.2

Multiply $u^{2}$ by $u^{2}$ by adding the exponents.

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Step 6.2.1

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

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

Step 6.2.2

Add $2$ and $2$ .

$\int u^{2} + u^{4} d u$

Step 7

Split the single integral into multiple integrals.

$\int u^{2} d u + \int u^{4} d u$

Step 8

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

$\frac{1}{3} u^{3} + C + \int u^{4} d u$

Step 9

By the Power Rule, the integral of $u^{4}$ with respect to $u$ is $\frac{1}{5} u^{5}$ .

$\frac{1}{3} u^{3} + C + \frac{1}{5} u^{5} + C$

Step 10

Simplify.

$\frac{1}{3} u^{3} + \frac{1}{5} u^{5} + C$

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

$\frac{1}{3} \tan^{3} (t) + \frac{1}{5} \tan^{5} (t) + C$

Step 11.2

Replace all occurrences of $t$ with $arcsec (x)$ .

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