Calculus Examples

\int x arcsin (x) d x

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

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.

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