Calculus Examples

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

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.

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

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.

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

Step 2.1.2

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

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

Step 2.2

Reduce the expression by cancelling the common factors.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factor.

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

Step 2.2.1.3

Rewrite the expression.

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

Step 2.2.2

Simplify.

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Step 2.2.2.1

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

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

Step 2.2.2.2

Rewrite $\frac{1^{2}}{\sec^{2} (t)}$ as ${(\frac{1}{\sec (t)})}^{2}$ .

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

Step 2.2.2.3

Rewrite $\sec (t)$ in terms of sines and cosines.

$\int {(\frac{1}{\frac{1}{\cos (t)}})}^{2} d t$

Step 2.2.2.4

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (t)}$ .

$\int {(1 \cos (t))}^{2} d t$

Step 2.2.2.5

Multiply $\cos (t)$ by $1$ .

$\int \cos^{2} (t) d t$

Step 3

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$\int \frac{1 + \cos (2 t)}{2} d t$

Step 4

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

$\frac{1}{2} \int 1 + \cos (2 t) d t$

Step 5

Split the single integral into multiple integrals.

$\frac{1}{2} (\int d t + \int \cos (2 t) d t)$

Step 6

Apply the constant rule.

$\frac{1}{2} (t + C + \int \cos (2 t) d t)$

Step 7

Let $u = 2 t$ . Then $d u = 2 d t$ , so $\frac{1}{2} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 7.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 7.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 7.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 7.1.3

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

$2 \cdot 1$

Step 7.1.4

Multiply $2$ by $1$ .

$2$

Step 7.2

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

$\frac{1}{2} (t + C + \int \cos (u) \frac{1}{2} d u)$

Step 8

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{1}{2} (t + C + \int \frac{\cos (u)}{2} d u)$

Step 9

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

$\frac{1}{2} (t + C + \frac{1}{2} \int \cos (u) d u)$

Step 10

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{1}{2} (t + C + \frac{1}{2} (\sin (u) + C))$

Step 11

Simplify.

$\frac{1}{2} (t + \frac{1}{2} \sin (u)) + C$

Step 12

Substitute back in for each integration substitution variable.

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Step 12.1

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

$\frac{1}{2} (arcsec (x) + \frac{1}{2} \sin (u)) + C$

Step 12.2

Replace all occurrences of $u$ with $2 t$ .

$\frac{1}{2} (arcsec (x) + \frac{1}{2} \sin (2 t)) + C$

Step 12.3

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

$\frac{1}{2} (arcsec (x) + \frac{1}{2} \sin (2 arcsec (x))) + C$

Step 13

Simplify.

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Step 13.1

Combine $\frac{1}{2}$ and $\sin (2 arcsec (x))$ .

$\frac{1}{2} (arcsec (x) + \frac{\sin (2 arcsec (x))}{2}) + C$

Step 13.2

Apply the distributive property.

$\frac{1}{2} arcsec (x) + \frac{1}{2} \cdot \frac{\sin (2 arcsec (x))}{2} + C$

Step 13.3

Combine $\frac{1}{2}$ and $arcsec (x)$ .

$\frac{arcsec (x)}{2} + \frac{1}{2} \cdot \frac{\sin (2 arcsec (x))}{2} + C$

Step 13.4

Multiply $\frac{1}{2} \cdot \frac{\sin (2 arcsec (x))}{2}$ .

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Step 13.4.1

Multiply $\frac{1}{2}$ by $\frac{\sin (2 arcsec (x))}{2}$ .

$\frac{arcsec (x)}{2} + \frac{\sin (2 arcsec (x))}{2 \cdot 2} + C$

Step 13.4.2

Multiply $2$ by $2$ .

$\frac{arcsec (x)}{2} + \frac{\sin (2 arcsec (x))}{4} + C$

Step 14

Reorder terms.

$\frac{1}{2} arcsec (x) + \frac{1}{4} \sin (2 arcsec (x)) + C$