Calculus Examples

$\int \arcsin (3 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \arcsin (3 x)$ and $d v = 1$ .

$\arcsin (3 x) x - \int x \frac{3}{\sqrt{1 - 9 x^{2}}} d x$

Step 2

Simplify.

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Step 2.1

Combine $x$ and $\frac{3}{\sqrt{1 - 9 x^{2}}}$ .

$\arcsin (3 x) x - \int \frac{x \cdot 3}{\sqrt{1 - 9 x^{2}}} d x$

Step 2.2

Move $3$ to the left of $x$ .

$\arcsin (3 x) x - \int \frac{3 x}{\sqrt{1 - 9 x^{2}}} d x$

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$\arcsin (3 x) x - (3 \int \frac{x}{\sqrt{1 - 9 x^{2}}} d x)$

Step 4

Multiply $3$ by $- 1$ .

$\arcsin (3 x) x - 3 \int \frac{x}{\sqrt{1 - 9 x^{2}}} d x$

Step 5

Let $u = 1 - 9 x^{2}$ . Then $d u = - 18 x d x$ , so $- \frac{1}{18} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 1 - 9 x^{2}$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $1 - 9 x^{2}$ .

$\frac{d}{d x} [1 - 9 x^{2}]$

Step 5.1.2

Differentiate.

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Step 5.1.2.1

By the Sum Rule, the derivative of $1 - 9 x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 9 x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- 9 x^{2}]$

Step 5.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 9 x^{2}]$

Step 5.1.3

Evaluate $\frac{d}{d x} [- 9 x^{2}]$ .

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Step 5.1.3.1

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9 x^{2}$ with respect to $x$ is $- 9 \frac{d}{d x} [x^{2}]$ .

$0 - 9 \frac{d}{d x} [x^{2}]$

Step 5.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - 9 (2 x)$

Step 5.1.3.3

Multiply $2$ by $- 9$ .

$0 - 18 x$

Step 5.1.4

Subtract $18 x$ from $0$ .

$- 18 x$

Step 5.2

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

$\arcsin (3 x) x - 3 \int \frac{1}{\sqrt{u}} \cdot \frac{1}{- 18} d u$

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

$\arcsin (3 x) x - 3 \int \frac{1}{\sqrt{u}} (- \frac{1}{18}) d u$

Step 6.2

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{18}$ .

$\arcsin (3 x) x - 3 \int - \frac{1}{\sqrt{u} \cdot 18} d u$

Step 6.3

Move $18$ to the left of $\sqrt{u}$ .

$\arcsin (3 x) x - 3 \int - \frac{1}{18 \sqrt{u}} d u$

Step 7

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

$\arcsin (3 x) x - 3 (- \int \frac{1}{18 \sqrt{u}} d u)$

Step 8

Multiply $- 1$ by $- 3$ .

$\arcsin (3 x) x + 3 \int \frac{1}{18 \sqrt{u}} d u$

Step 9

Since $\frac{1}{18}$ is constant with respect to $u$ , move $\frac{1}{18}$ out of the integral.

$\arcsin (3 x) x + 3 (\frac{1}{18} \int \frac{1}{\sqrt{u}} d u)$

Step 10

Simplify the expression.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{1}{18}$ and $3$ .

$\arcsin (3 x) x + \frac{3}{18} \int \frac{1}{\sqrt{u}} d u$

Step 10.1.2

Cancel the common factor of $3$ and $18$ .

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Step 10.1.2.1

Factor $3$ out of $3$ .

$\arcsin (3 x) x + \frac{3 (1)}{18} \int \frac{1}{\sqrt{u}} d u$

Step 10.1.2.2

Cancel the common factors.

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Step 10.1.2.2.1

Factor $3$ out of $18$ .

$\arcsin (3 x) x + \frac{3 \cdot 1}{3 \cdot 6} \int \frac{1}{\sqrt{u}} d u$

Step 10.1.2.2.2

Cancel the common factor.

$\arcsin (3 x) x + \frac{3 \cdot 1}{3 \cdot 6} \int \frac{1}{\sqrt{u}} d u$

Step 10.1.2.2.3

Rewrite the expression.

$\arcsin (3 x) x + \frac{1}{6} \int \frac{1}{\sqrt{u}} d u$

Step 10.2

Apply basic rules of exponents.

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Step 10.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\arcsin (3 x) x + \frac{1}{6} \int \frac{1}{u^{\frac{1}{2}}} d u$

Step 10.2.2

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

$\arcsin (3 x) x + \frac{1}{6} \int {(u^{\frac{1}{2}})}^{- 1} d u$

Step 10.2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 10.2.3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\arcsin (3 x) x + \frac{1}{6} \int u^{\frac{1}{2} \cdot - 1} d u$

Step 10.2.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$\arcsin (3 x) x + \frac{1}{6} \int u^{\frac{- 1}{2}} d u$

Step 10.2.3.3

Move the negative in front of the fraction.

$\arcsin (3 x) x + \frac{1}{6} \int u^{- \frac{1}{2}} d u$

Step 11

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

$\arcsin (3 x) x + \frac{1}{6} (2 u^{\frac{1}{2}} + C)$

Step 12

Rewrite $\arcsin (3 x) x + \frac{1}{6} (2 u^{\frac{1}{2}} + C)$ as $\arcsin (3 x) x + \frac{u^{\frac{1}{2}}}{3} + C$ .

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

Step 13

Replace all occurrences of $u$ with $1 - 9 x^{2}$ .

$\arcsin (3 x) x + \frac{{(1 - 9 x^{2})}^{\frac{1}{2}}}{3} + C$

Step 14

Reorder terms.

$\arcsin (3 x) x + \frac{1}{3} {(1 - 9 x^{2})}^{\frac{1}{2}} + C$