Calculus Examples

\arcsin (x) d x

$\arcsin (x) d x$

Step 1

Integrate by parts using the formula

\int u d v = u v - \int v d u

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

u = \arcsin (x)

$u = \arcsin (x)$ and

d v = 1

$d v = 1$ .

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

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

Step 2

Combine

x

$x$ and

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

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

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

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

Step 3

Let

u = 1 - x^{2}

$u = 1 - x^{2}$ . Then

d u = - 2 x d x

$d u = - 2 x d x$ , so

- \frac{1}{2} d u = x d x

$- \frac{1}{2} d u = x d x$ . Rewrite using

u

$u$ and

d

$d$

u

$u$ .

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Step 3.1

Let

u = 1 - x^{2}

$u = 1 - x^{2}$ . Find

\frac{d u}{d x}

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

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Step 3.1.1

Differentiate

1 - x^{2}

$1 - x^{2}$ .

\frac{d}{d x} [1 - x^{2}]

$\frac{d}{d x} [1 - x^{2}]$

Step 3.1.2

Differentiate.

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Step 3.1.2.1

By the Sum Rule, the derivative of

1 - x^{2}

$1 - x^{2}$ with respect to

x

$x$ is

\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]

$\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$ .

\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]

$\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$

Step 3.1.2.2

Since

1

$1$ is constant with respect to

x

$x$ , the derivative of

1

$1$ with respect to

x

$x$ is

0

$0$ .

0 + \frac{d}{d x} [- x^{2}]

$0 + \frac{d}{d x} [- x^{2}]$

0 + \frac{d}{d x} [- x^{2}]

$0 + \frac{d}{d x} [- x^{2}]$

Step 3.1.3

Evaluate

\frac{d}{d x} [- x^{2}]

$\frac{d}{d x} [- x^{2}]$ .

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Step 3.1.3.1

Since

- 1

$- 1$ is constant with respect to

x

$x$ , the derivative of

- x^{2}

$- x^{2}$ with respect to

x

$x$ is

- \frac{d}{d x} [x^{2}]

$- \frac{d}{d x} [x^{2}]$ .

0 - \frac{d}{d x} [x^{2}]

$0 - \frac{d}{d x} [x^{2}]$

Step 3.1.3.2

Differentiate using the Power Rule which states that

\frac{d}{d x} [x^{n}]

$\frac{d}{d x} [x^{n}]$ is

n x^{n - 1}

$n x^{n - 1}$ where

n = 2

$n = 2$ .

0 - (2 x)

$0 - (2 x)$

Step 3.1.3.3

Multiply

2

$2$ by

- 1

$- 1$ .

0 - 2 x

$0 - 2 x$

0 - 2 x

$0 - 2 x$

Step 3.1.4

Subtract

2 x

$2 x$ from

0

$0$ .

- 2 x

$- 2 x$

- 2 x

$- 2 x$

Step 3.2

Rewrite the problem using

u

$u$ and

d u

$d u$ .

\arcsin (x) x - \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u

$\arcsin (x) x - \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

\arcsin (x) x - \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u

$\arcsin (x) x - \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

\arcsin (x) x - \int \frac{1}{\sqrt{u}} (- \frac{1}{2}) d u

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

Step 4.2

Multiply

\frac{1}{\sqrt{u}}

$\frac{1}{\sqrt{u}}$ by

\frac{1}{2}

$\frac{1}{2}$ .

\arcsin (x) x - \int - \frac{1}{\sqrt{u} \cdot 2} d u

$\arcsin (x) x - \int - \frac{1}{\sqrt{u} \cdot 2} d u$

Step 4.3

Move

2

$2$ to the left of

\sqrt{u}

$\sqrt{u}$ .

\arcsin (x) x - \int - \frac{1}{2 \sqrt{u}} d u

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

\arcsin (x) x - \int - \frac{1}{2 \sqrt{u}} d u

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

Step 5

Since

- 1

$- 1$ is constant with respect to

u

$u$ , move

- 1

$- 1$ out of the integral.

\arcsin (x) x - - \int \frac{1}{2 \sqrt{u}} d u

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

Step 6

Simplify.

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Step 6.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

\arcsin (x) x + 1 \int \frac{1}{2 \sqrt{u}} d u

$\arcsin (x) x + 1 \int \frac{1}{2 \sqrt{u}} d u$

Step 6.2

Multiply

\int \frac{1}{2 \sqrt{u}} d u

$\int \frac{1}{2 \sqrt{u}} d u$ by

1

$1$ .

\arcsin (x) x + \int \frac{1}{2 \sqrt{u}} d u

$\arcsin (x) x + \int \frac{1}{2 \sqrt{u}} d u$

\arcsin (x) x + \int \frac{1}{2 \sqrt{u}} d u

$\arcsin (x) x + \int \frac{1}{2 \sqrt{u}} d u$

Step 7

Since

\frac{1}{2}

$\frac{1}{2}$ is constant with respect to

u

$u$ , move

\frac{1}{2}

$\frac{1}{2}$ out of the integral.

\arcsin (x) x + \frac{1}{2} \int \frac{1}{\sqrt{u}} d u

$\arcsin (x) x + \frac{1}{2} \int \frac{1}{\sqrt{u}} d u$

Step 8

Apply basic rules of exponents.

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Step 8.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{u}

$\sqrt{u}$ as

u^{\frac{1}{2}}

$u^{\frac{1}{2}}$ .

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

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

Step 8.2

Move

u^{\frac{1}{2}}

$u^{\frac{1}{2}}$ out of the denominator by raising it to the

- 1

$- 1$ power.

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

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

Step 8.3

Multiply the exponents in

{(u^{\frac{1}{2}})}^{- 1}

${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 8.3.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 8.3.2

Combine

\frac{1}{2}

$\frac{1}{2}$ and

- 1

$- 1$ .

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

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

Step 8.3.3

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 9

By the Power Rule, the integral of

u^{- \frac{1}{2}}

$u^{- \frac{1}{2}}$ with respect to

u

$u$ is

2 u^{\frac{1}{2}}

$2 u^{\frac{1}{2}}$ .

\arcsin (x) x + \frac{1}{2} (2 u^{\frac{1}{2}} + C)

$\arcsin (x) x + \frac{1}{2} (2 u^{\frac{1}{2}} + C)$

Step 10

Rewrite

\arcsin (x) x + \frac{1}{2} (2 u^{\frac{1}{2}} + C)

$\arcsin (x) x + \frac{1}{2} (2 u^{\frac{1}{2}} + C)$ as

\arcsin (x) x + u^{\frac{1}{2}} + C

$\arcsin (x) x + u^{\frac{1}{2}} + C$ .

\arcsin (x) x + u^{\frac{1}{2}} + C

$\arcsin (x) x + u^{\frac{1}{2}} + C$

Step 11

Replace all occurrences of

u

$u$ with

1 - x^{2}

$1 - x^{2}$ .

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

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