Calculus Examples

$\int \cos^{-1} (x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \cos^{-1} (x)$ and $d v = 1$ .

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

Multiply $\int \frac{x}{\sqrt{1 - x^{2}}} d x$ by $1$ .

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

Step 5

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

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Step 5.1

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

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Step 5.1.1

Differentiate $1 - x^{2}$ .

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

Step 5.1.2

Differentiate.

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Step 5.1.2.1

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

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

Step 5.1.2.2

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

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

Step 5.1.3

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

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Step 5.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

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 $n = 2$ .

$0 - (2 x)$

Step 5.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 5.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 5.2

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

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

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

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

Step 6.2

Multiply $\frac{1}{\sqrt{u}}$ by $\frac{1}{2}$ .

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

Step 6.3

Move $2$ to the left of $\sqrt{u}$ .

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

Step 7

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

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

Step 8

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 9

Apply basic rules of exponents.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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Step 9.3.1

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

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

Step 9.3.2

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

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

Step 9.3.3

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

Rewrite $\cos^{-1} (x) x - \frac{1}{2} (2 u^{\frac{1}{2}} + C)$ as $\cos^{-1} (x) x - u^{\frac{1}{2}} + C$ .

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

Step 12

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

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