Calculus Examples

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

Step 1

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

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

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.

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

Step 4

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

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

Step 5

Simplify terms.

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Step 5.1

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

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Step 5.1.1

Apply pythagorean identity.

$\int \frac{\sin^{3} (t)}{\sqrt{\cos^{2} (t)}} \cos (t) d t$

Step 5.1.2

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

$\int \frac{\sin^{3} (t)}{\cos (t)} \cos (t) d t$

Step 5.2

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

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Step 5.2.1

Cancel the common factor.

$\int \frac{\sin^{3} (t)}{\cos (t)} \cos (t) d t$

Step 5.2.2

Rewrite the expression.

$\int \sin^{3} (t) d t$

Step 6

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

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

Step 7

Using the Pythagorean Identity, rewrite $\sin^{2} (t)$ as $1 - \cos^{2} (t)$ .

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

Step 8

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

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Step 8.1

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

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Step 8.1.1

Differentiate $\cos (t)$ .

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

Step 8.1.2

The derivative of $\cos (t)$ with respect to $t$ is $- \sin (t)$ .

$- \sin (t)$

Step 8.2

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

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

Step 9

Split the single integral into multiple integrals.

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

Step 10

Apply the constant rule.

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

Step 11

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 12

Simplify.

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

Step 13

Substitute back in for each integration substitution variable.

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Step 13.1

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

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

Step 13.2

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

$- \cos (\arcsin (x)) + \frac{1}{3} \cos^{3} (\arcsin (x)) + C$

Step 14

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

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