Calculus Examples

$\frac{x^{3}}{\sqrt{x^{2} + 1}}$

Step 1

Write $\frac{x^{3}}{\sqrt{x^{2} + 1}}$ as a function.

$f (x) = \frac{x^{3}}{\sqrt{x^{2} + 1}}$

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.

$F (x) = \int \frac{x^{3}}{\sqrt{x^{2} + 1}} d x$

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

Simplify $\sqrt{\tan^{2} (t) + 1}$ .

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Step 5.1.1

Apply pythagorean identity.

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

Step 5.1.2

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

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

Step 5.2

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

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Step 5.2.1

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

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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

Step 10

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

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Step 10.1

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

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Step 10.1.1

Differentiate $\sec (t)$ .

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

Step 10.1.2

The derivative of $\sec (t)$ with respect to $t$ is $\sec (t) \tan (t)$ .

$\sec (t) \tan (t)$

Step 10.2

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

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

Step 11

Split the single integral into multiple integrals.

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

Step 12

Apply the constant rule.

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

Step 13

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

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

Step 14

Simplify.

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

Step 15

Substitute back in for each integration substitution variable.

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Step 15.1

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

$- \sec (t) + \frac{1}{3} \sec^{3} (t) + C$

Step 15.2

Replace all occurrences of $t$ with $\arctan (x)$ .

$- \sec (\arctan (x)) + \frac{1}{3} \sec^{3} (\arctan (x)) + C$

Step 16

The answer is the antiderivative of the function $f (x) = \frac{x^{3}}{\sqrt{x^{2} + 1}}$ .

$F (x) =$ $- \sec (\arctan (x)) + \frac{1}{3} \sec^{3} (\arctan (x)) + C$