Calculus Examples

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

Step 1

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

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

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.

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

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.

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

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.

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

Step 5.1.2

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

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

Step 5.2

Simplify.

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Step 5.2.1

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

$\int \sin^{3} (t) (\cos^{1} (t) \cos (t)) d t$

Step 5.2.2

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

$\int \sin^{3} (t) (\cos^{1} (t) \cos^{1} (t)) d t$

Step 5.2.3

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

$\int \sin^{3} (t) {\cos (t)}^{1 + 1} d t$

Step 5.2.4

Add $1$ and $1$ .

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

Step 6

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

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

Step 7

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

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

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 $d u$ .

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

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

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Step 10.2.1

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

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

Step 10.2.2

Add $2$ and $2$ .

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

Step 11

Split the single integral into multiple integrals.

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

Step 12

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

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

Step 13

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 14

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 15

Simplify.

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Step 15.1

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

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

Step 15.2

Simplify.

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

Step 16

Substitute back in for each integration substitution variable.

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Step 16.1

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

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

Step 16.2

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

$- \frac{1}{3} \cos^{3} (\arcsin (x)) + \frac{1}{5} \cos^{5} (\arcsin (x)) + C$

Step 17

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

$F (x) =$ $- \frac{1}{3} \cos^{3} (\arcsin (x)) + \frac{1}{5} \cos^{5} (\arcsin (x)) + C$