Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

Combine $\arcsin (x)$ and $\frac{x^{2}}{2}$ .

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

Step 3

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

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

Step 4

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

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

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.

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

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.

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

Step 6.1.2

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

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

Step 6.2

Cancel the common factor of $\cos (t)$ .

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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

Step 7

Use the half-angle formula to rewrite $\sin^{2} (t)$ as $\frac{1 - \cos (2 t)}{2}$ .

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Multiply $2$ by $2$ .

$\frac{\arcsin (x) x^{2}}{2} - \frac{1}{4} \int 1 - \cos (2 t) d t$

Step 10

Split the single integral into multiple integrals.

$\frac{\arcsin (x) x^{2}}{2} - \frac{1}{4} (\int d t + \int - \cos (2 t) d t)$

Step 11

Apply the constant rule.

$\frac{\arcsin (x) x^{2}}{2} - \frac{1}{4} (t + C + \int - \cos (2 t) d t)$

Step 12

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

$\frac{\arcsin (x) x^{2}}{2} - \frac{1}{4} (t + C - \int \cos (2 t) d t)$

Step 13

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

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Step 13.1

Let $u = 2 t$ . Find $\frac{d u}{d t}$ .

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Step 13.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 13.1.2

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

$2 \frac{d}{d t} [t]$

Step 13.1.3

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

$2 \cdot 1$

Step 13.1.4

Multiply $2$ by $1$ .

$2$

Step 13.2

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

$\frac{\arcsin (x) x^{2}}{2} - \frac{1}{4} (t + C - \int \cos (u) \frac{1}{2} d u)$

Step 14

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{\arcsin (x) x^{2}}{2} - \frac{1}{4} (t + C - \int \frac{\cos (u)}{2} d u)$

Step 15

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

$\frac{\arcsin (x) x^{2}}{2} - \frac{1}{4} (t + C - (\frac{1}{2} \int \cos (u) d u))$

Step 16

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$\frac{\arcsin (x) x^{2}}{2} - \frac{1}{4} (t + C - \frac{1}{2} (\sin (u) + C))$

Step 17

Simplify.

$\frac{1}{2} \arcsin (x) x^{2} - \frac{1}{4} (t - \frac{1}{2} \sin (u)) + C$

Step 18

Substitute back in for each integration substitution variable.

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Step 18.1

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

$\frac{1}{2} \arcsin (x) x^{2} - \frac{1}{4} (\arcsin (x) - \frac{1}{2} \sin (u)) + C$

Step 18.2

Replace all occurrences of $u$ with $2 t$ .

$\frac{1}{2} \arcsin (x) x^{2} - \frac{1}{4} (\arcsin (x) - \frac{1}{2} \sin (2 t)) + C$

Step 18.3

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

$\frac{1}{2} \arcsin (x) x^{2} - \frac{1}{4} (\arcsin (x) - \frac{1}{2} \sin (2 \arcsin (x))) + C$

Step 19

Simplify.

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Step 19.1

Combine $\sin (2 \arcsin (x))$ and $\frac{1}{2}$ .

$\frac{1}{2} \arcsin (x) x^{2} - \frac{1}{4} (\arcsin (x) - \frac{\sin (2 \arcsin (x))}{2}) + C$

Step 19.2

Apply the distributive property.

$\frac{1}{2} \arcsin (x) x^{2} - \frac{1}{4} \arcsin (x) - \frac{1}{4} (- \frac{\sin (2 \arcsin (x))}{2}) + C$

Step 19.3

Combine $\arcsin (x)$ and $\frac{1}{4}$ .

$\frac{1}{2} \arcsin (x) x^{2} - \frac{\arcsin (x)}{4} - \frac{1}{4} (- \frac{\sin (2 \arcsin (x))}{2}) + C$

Step 19.4

Multiply $- \frac{1}{4} (- \frac{\sin (2 \arcsin (x))}{2})$ .

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Step 19.4.1

Multiply $- 1$ by $- 1$ .

$\frac{1}{2} \arcsin (x) x^{2} - \frac{\arcsin (x)}{4} + 1 (\frac{1}{4}) \frac{\sin (2 \arcsin (x))}{2} + C$

Step 19.4.2

Multiply $\frac{1}{4}$ by $1$ .

$\frac{1}{2} \arcsin (x) x^{2} - \frac{\arcsin (x)}{4} + \frac{1}{4} \cdot \frac{\sin (2 \arcsin (x))}{2} + C$

Step 19.4.3

Multiply $\frac{1}{4}$ by $\frac{\sin (2 \arcsin (x))}{2}$ .

$\frac{1}{2} \arcsin (x) x^{2} - \frac{\arcsin (x)}{4} + \frac{\sin (2 \arcsin (x))}{4 \cdot 2} + C$

Step 19.4.4

Multiply $4$ by $2$ .

$\frac{1}{2} \arcsin (x) x^{2} - \frac{\arcsin (x)}{4} + \frac{\sin (2 \arcsin (x))}{8} + C$

Step 19.5

Combine $\frac{1}{2}$ and $\arcsin (x)$ .

$\frac{\arcsin (x)}{2} x^{2} - \frac{\arcsin (x)}{4} + \frac{\sin (2 \arcsin (x))}{8} + C$

Step 19.6

Combine $\frac{\arcsin (x)}{2}$ and $x^{2}$ .

$\frac{\arcsin (x) x^{2}}{2} - \frac{\arcsin (x)}{4} + \frac{\sin (2 \arcsin (x))}{8} + C$

Step 20

Simplify.

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Step 20.1

Reorder factors in $\frac{\arcsin (x) x^{2}}{2}$ .

$\frac{x^{2} \arcsin (x)}{2} - \frac{\arcsin (x)}{4} + \frac{\sin (2 \arcsin (x))}{8} + C$

Step 20.2

Reorder terms.

$\frac{1}{2} x^{2} \arcsin (x) - \frac{1}{4} \arcsin (x) + \frac{1}{8} \sin (2 \arcsin (x)) + C$