Calculus Examples

$\int x^{2} \arcsin (x) d x$

Step 1

Integrate by parts using the formula

$\int u d v = u v - \int v d u$ , where

$u = \arcsin (x)$ and

$d v = x^{2}$ .

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

Step 2

Simplify.

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Step 2.1

Combine

$\frac{1}{3}$ and

$x^{3}$ .

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

Step 2.2

Combine

$\arcsin (x)$ and

$\frac{x^{3}}{3}$ .

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

Step 3

Since

$\frac{1}{3}$ is constant with respect to

$x$ , move

$\frac{1}{3}$ out of the integral.

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

Step 4

Combine

$x^{3}$ and

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

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

Step 5

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.

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

Step 6

Simplify terms.

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Step 6.1

Simplify

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

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Step 6.1.1

Apply pythagorean identity.

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

Step 6.1.2

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

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

Step 6.2

Cancel the common factor of

$\cos (t)$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

$\frac{\arcsin (x) x^{3}}{3} - \frac{1}{3} \int \sin^{3} (t) d t$

Step 7

Factor out

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

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

Step 8

Using the Pythagorean Identity, rewrite

$\sin^{2} (t)$ as

$1 - \cos^{2} (t)$ .

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

Step 9

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 9.1

Let

$u = \cos (t)$ . Find

$\frac{d u}{d t}$ .

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Step 9.1.1

Differentiate

$\cos (t)$ .

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

Step 9.1.2

The derivative of

$\cos (t)$ with respect to

$t$ is

$- \sin (t)$ .

$- \sin (t)$

Step 9.2

Rewrite the problem using

$u$ and

$d u$ .

$\frac{\arcsin (x) x^{3}}{3} - \frac{1}{3} \int - 1 + u^{2} d u$

Step 10

Split the single integral into multiple integrals.

$\frac{\arcsin (x) x^{3}}{3} - \frac{1}{3} (\int - 1 d u + \int u^{2} d u)$

Step 11

Apply the constant rule.

$\frac{\arcsin (x) x^{3}}{3} - \frac{1}{3} (- u + C + \int u^{2} d u)$

Step 12

By the Power Rule, the integral of

$u^{2}$ with respect to

$u$ is

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

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

Step 13

Simplify.

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Step 13.1

Simplify.

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

Step 13.2

Simplify.

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Step 13.2.1

Combine

$\frac{1}{3}$ and

$\arcsin (x)$ .

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

Step 13.2.2

Combine

$\frac{\arcsin (x)}{3}$ and

$x^{3}$ .

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

Step 13.2.3

To write

$- \frac{1}{3} (- u + \frac{1}{3} u^{3})$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

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

Step 13.2.4

Combine

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

$\frac{3}{3}$ .

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

Step 13.2.5

Combine the numerators over the common denominator.

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

Step 13.2.6

Combine

$\frac{1}{3}$ and

$u^{3}$ .

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

Step 13.2.7

Multiply

$3$ by

$- 1$ .

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

Step 13.2.8

Combine

$- 3$ and

$\frac{1}{3}$ .

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

Step 13.2.9

Cancel the common factor of

$- 3$ and

$3$ .

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Step 13.2.9.1

Factor

$3$ out of

$- 3$ .

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

Step 13.2.9.2

Cancel the common factors.

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Step 13.2.9.2.1

Factor

$3$ out of

$3$ .

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

Step 13.2.9.2.2

Cancel the common factor.

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

Step 13.2.9.2.3

Rewrite the expression.

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

Step 13.2.9.2.4

Divide

$- 1$ by

$1$ .

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

Step 14

Substitute back in for each integration substitution variable.

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Step 14.1

Replace all occurrences of

$u$ with

$\cos (t)$ .

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

Step 14.2

Replace all occurrences of

$t$ with

$\arcsin (x)$ .

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

Step 15

Simplify.

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Step 15.1

Simplify each term.

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Step 15.1.1

Draw a triangle in the plane with vertices

$(\sqrt{1^{2} - x^{2}}, x)$ ,

$(\sqrt{1^{2} - x^{2}}, 0)$ , and the origin. Then

$\arcsin (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

$(\sqrt{1^{2} - x^{2}}, x)$ . Therefore,

$\cos (\arcsin (x))$ is

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

$\frac{\arcsin (x) x^{3} - (- \sqrt{1 - x^{2}} + \frac{\cos^{3} (\arcsin (x))}{3})}{3} + C$

Step 15.1.2

Rewrite

$1$ as

$1^{2}$ .

$\frac{\arcsin (x) x^{3} - (- \sqrt{1^{2} - x^{2}} + \frac{\cos^{3} (\arcsin (x))}{3})}{3} + C$

Step 15.1.3

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 1$ and

$b = x$ .

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

Step 15.2

Apply the distributive property.

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

Step 15.3

Multiply

$- - \sqrt{(1 + x) (1 - x)}$ .

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Step 15.3.1

Multiply

$- 1$ by

$- 1$ .

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

Step 15.3.2

Multiply

$\sqrt{(1 + x) (1 - x)}$ by

$1$ .

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

Step 15.4

Simplify the numerator.

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Step 15.4.1

Draw a triangle in the plane with vertices

$(\sqrt{1^{2} - x^{2}}, x)$ ,

$(\sqrt{1^{2} - x^{2}}, 0)$ , and the origin. Then

$\arcsin (x)$ is the angle between the positive x-axis and the ray beginning at the origin and passing through

$(\sqrt{1^{2} - x^{2}}, x)$ . Therefore,

$\cos (\arcsin (x))$ is

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

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{{\sqrt{1 - x^{2}}}^{3}}{3}}{3} + C$

Step 15.4.2

Rewrite

$1$ as

$1^{2}$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{{\sqrt{1^{2} - x^{2}}}^{3}}{3}}{3} + C$

Step 15.4.3

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 1$ and

$b = x$ .

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

Step 15.4.4

Rewrite

${\sqrt{(1 + x) (1 - x)}}^{3}$ as

$\sqrt{{((1 + x) (1 - x))}^{3}}$ .

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

Step 15.4.5

Apply the product rule to

$(1 + x) (1 - x)$ .

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

Step 15.4.6

Rewrite

${(1 + x)}^{3} {(1 - x)}^{3}$ as

${((1 + x) (1 - x))}^{2} ((1 + x) (1 - x))$ .

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Step 15.4.6.1

Factor out

${(1 + x)}^{2}$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{{(1 + x)}^{2} (1 + x) {(1 - x)}^{3}}}{3}}{3} + C$

Step 15.4.6.2

Factor out

${(1 - x)}^{2}$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{{(1 + x)}^{2} (1 + x) ({(1 - x)}^{2} (1 - x))}}{3}}{3} + C$

Step 15.4.6.3

Move

$1 + x$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{{(1 + x)}^{2} {(1 - x)}^{2} (1 + x) (1 - x)}}{3}}{3} + C$

Step 15.4.6.4

Rewrite

${(1 + x)}^{2} {(1 - x)}^{2}$ as

${((1 + x) (1 - x))}^{2}$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{{((1 + x) (1 - x))}^{2} (1 + x) (1 - x)}}{3}}{3} + C$

Step 15.4.6.5

Add parentheses.

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{{((1 + x) (1 - x))}^{2} ((1 + x) (1 - x))}}{3}}{3} + C$

Step 15.4.7

Pull terms out from under the radical.

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

Step 15.4.8

Expand

$(1 + x) (1 - x)$ using the FOIL Method.

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Step 15.4.8.1

Apply the distributive property.

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

Step 15.4.8.2

Apply the distributive property.

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

Step 15.4.8.3

Apply the distributive property.

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

Step 15.4.9

Simplify and combine like terms.

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Step 15.4.9.1

Simplify each term.

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Step 15.4.9.1.1

Multiply

$1$ by

$1$ .

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

Step 15.4.9.1.2

Multiply

$- x$ by

$1$ .

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

Step 15.4.9.1.3

Multiply

$x$ by

$1$ .

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

Step 15.4.9.1.4

Rewrite using the commutative property of multiplication.

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

Step 15.4.9.1.5

Multiply

$x$ by

$x$ by adding the exponents.

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Step 15.4.9.1.5.1

Move

$x$ .

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

Step 15.4.9.1.5.2

Multiply

$x$ by

$x$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{(1 - x + x - x^{2}) \sqrt{(1 + x) (1 - x)}}{3}}{3} + C$

Step 15.4.9.2

Add

$- x$ and

$x$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{(1 + 0 - x^{2}) \sqrt{(1 + x) (1 - x)}}{3}}{3} + C$

Step 15.4.9.3

Add

$1$ and

$0$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{(1 - x^{2}) \sqrt{(1 + x) (1 - x)}}{3}}{3} + C$

Step 15.4.10

Apply the distributive property.

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{1 \sqrt{(1 + x) (1 - x)} - x^{2} \sqrt{(1 + x) (1 - x)}}{3}}{3} + C$

Step 15.4.11

Multiply

$\sqrt{(1 + x) (1 - x)}$ by

$1$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{(1 + x) (1 - x)} - x^{2} \sqrt{(1 + x) (1 - x)}}{3}}{3} + C$

Step 15.4.12

Factor

$\sqrt{(1 + x) (1 - x)}$ out of

$\sqrt{(1 + x) (1 - x)} - x^{2} \sqrt{(1 + x) (1 - x)}$ .

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Step 15.4.12.1

Multiply by

$1$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{(1 + x) (1 - x)} \cdot 1 - x^{2} \sqrt{(1 + x) (1 - x)}}{3}}{3} + C$

Step 15.4.12.2

Factor

$\sqrt{(1 + x) (1 - x)}$ out of

$- x^{2} \sqrt{(1 + x) (1 - x)}$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{(1 + x) (1 - x)} \cdot 1 + \sqrt{(1 + x) (1 - x)} (- x^{2})}{3}}{3} + C$

Step 15.4.12.3

Factor

$\sqrt{(1 + x) (1 - x)}$ out of

$\sqrt{(1 + x) (1 - x)} \cdot 1 + \sqrt{(1 + x) (1 - x)} (- x^{2})$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{(1 + x) (1 - x)} (1 - x^{2})}{3}}{3} + C$

Step 15.4.13

Rewrite

$1$ as

$1^{2}$ .

$\frac{\arcsin (x) x^{3} + \sqrt{(1 + x) (1 - x)} - \frac{\sqrt{(1 + x) (1 - x)} (1^{2} - x^{2})}{3}}{3} + C$

Step 15.4.14

Since both terms are perfect squares, factor using the difference of squares formula,

$a^{2} - b^{2} = (a + b) (a - b)$ where

$a = 1$ and

$b = x$ .

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

Step 15.5

To write

$\sqrt{(1 + x) (1 - x)}$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

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

Step 15.6

Combine

$\sqrt{(1 + x) (1 - x)}$ and

$\frac{3}{3}$ .

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

Step 15.7

Combine the numerators over the common denominator.

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

Step 15.8

Simplify the numerator.

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Step 15.8.1

Factor

$\sqrt{(1 + x) (1 - x)}$ out of

$\sqrt{(1 + x) (1 - x)} \cdot 3 - \sqrt{(1 + x) (1 - x)} (1 + x) (1 - x)$ .

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Step 15.8.1.1

Factor

$\sqrt{(1 + x) (1 - x)}$ out of

$\sqrt{(1 + x) (1 - x)} \cdot 3$ .

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

Step 15.8.1.2

Factor

$\sqrt{(1 + x) (1 - x)}$ out of

$- \sqrt{(1 + x) (1 - x)} (1 + x) (1 - x)$ .

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

Step 15.8.1.3

Factor

$\sqrt{(1 + x) (1 - x)}$ out of

$\sqrt{(1 + x) (1 - x)} (3) + \sqrt{(1 + x) (1 - x)} ((- 1 (1 + x)) (1 - x))$ .

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

Step 15.8.2

Apply the distributive property.

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

Step 15.8.3

Multiply

$- 1$ by

$1$ .

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

Step 15.8.4

Rewrite

$- 1 x$ as

$- x$ .

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

Step 15.8.5

Expand

$(- 1 - x) (1 - x)$ using the FOIL Method.

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Step 15.8.5.1

Apply the distributive property.

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

Step 15.8.5.2

Apply the distributive property.

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

Step 15.8.5.3

Apply the distributive property.

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

Step 15.8.6

Simplify and combine like terms.

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Step 15.8.6.1

Simplify each term.

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Step 15.8.6.1.1

Multiply

$- 1$ by

$1$ .

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

Step 15.8.6.1.2

Multiply

$- 1 (- x)$ .

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Step 15.8.6.1.2.1

Multiply

$- 1$ by

$- 1$ .

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

Step 15.8.6.1.2.2

Multiply

$x$ by

$1$ .

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

Step 15.8.6.1.3

Multiply

$- 1$ by

$1$ .

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

Step 15.8.6.1.4

Rewrite using the commutative property of multiplication.

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

Step 15.8.6.1.5

Multiply

$x$ by

$x$ by adding the exponents.

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Step 15.8.6.1.5.1

Move

$x$ .

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

Step 15.8.6.1.5.2

Multiply

$x$ by

$x$ .

$\frac{\arcsin (x) x^{3} + \frac{\sqrt{(1 + x) (1 - x)} (3 - 1 + x - x - 1 \cdot - 1 x^{2})}{3}}{3} + C$

Step 15.8.6.1.6

Multiply

$- 1$ by

$- 1$ .

$\frac{\arcsin (x) x^{3} + \frac{\sqrt{(1 + x) (1 - x)} (3 - 1 + x - x + 1 x^{2})}{3}}{3} + C$

Step 15.8.6.1.7

Multiply

$x^{2}$ by

$1$ .

$\frac{\arcsin (x) x^{3} + \frac{\sqrt{(1 + x) (1 - x)} (3 - 1 + x - x + x^{2})}{3}}{3} + C$

Step 15.8.6.2

Subtract

$x$ from

$x$ .

$\frac{\arcsin (x) x^{3} + \frac{\sqrt{(1 + x) (1 - x)} (3 - 1 + 0 + x^{2})}{3}}{3} + C$

Step 15.8.6.3

Add

$- 1$ and

$0$ .

$\frac{\arcsin (x) x^{3} + \frac{\sqrt{(1 + x) (1 - x)} (3 - 1 + x^{2})}{3}}{3} + C$

Step 15.8.7

Subtract

$1$ from

$3$ .

$\frac{\arcsin (x) x^{3} + \frac{\sqrt{(1 + x) (1 - x)} (2 + x^{2})}{3}}{3} + C$

Step 16

Reorder terms.

$\frac{1}{3} (\arcsin (x) x^{3} + \frac{\sqrt{(1 + x) (1 - x)} (2 + x^{2})}{3}) + C$