Calculus Examples

$\int_{0}^{1} \arcsin (x) d x$

Step 1

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

$\arcsin (x) x]_{0}^{1} - \int_{0}^{1} x \frac{1}{\sqrt{1 - x^{2}}} d x$

Step 2

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

$\arcsin (x) x]_{0}^{1} - \int_{0}^{1} \frac{x}{\sqrt{1 - x^{2}}} d x$

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $1 - x^{2}$ .

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

Step 3.1.2

Differentiate.

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Step 3.1.2.1

By the Sum Rule, the derivative of $1 - x^{2}$ 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 3.1.2.2

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 3.1.3

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

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Step 3.1.3.1

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 3.1.3.2

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 3.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 3.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 3.2

Substitute the lower limit in for $x$ in $u = 1 - x^{2}$ .

$u_{lower} = 1 - 0^{2}$

Step 3.3

Simplify.

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Step 3.3.1

Simplify each term.

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Step 3.3.1.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 1 - 0$

Step 3.3.1.2

Multiply $- 1$ by $0$ .

$u_{lower} = 1 + 0$

Step 3.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 3.4

Substitute the upper limit in for $x$ in $u = 1 - x^{2}$ .

$u_{upper} = 1 - 1^{2}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify each term.

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Step 3.5.1.1

One to any power is one.

$u_{upper} = 1 - 1 \cdot 1$

Step 3.5.1.2

Multiply $- 1$ by $1$ .

$u_{upper} = 1 - 1$

Step 3.5.2

Subtract $1$ from $1$ .

$u_{upper} = 0$

Step 3.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 0$

Step 3.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\arcsin (x) x]_{0}^{1} - \int_{1}^{0} \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 4

Simplify.

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Step 4.1

Move the negative in front of the fraction.

$\arcsin (x) x]_{0}^{1} - \int_{1}^{0} \frac{1}{\sqrt{u}} (- \frac{1}{2}) d u$

Step 4.2

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

$\arcsin (x) x]_{0}^{1} - \int_{1}^{0} - \frac{1}{\sqrt{u} \cdot 2} d u$

Step 4.3

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

$\arcsin (x) x]_{0}^{1} - \int_{1}^{0} - \frac{1}{2 \sqrt{u}} d u$

Step 5

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

$\arcsin (x) x]_{0}^{1} - - \int_{1}^{0} \frac{1}{2 \sqrt{u}} d u$

Step 6

Simplify.

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Step 6.1

Multiply $- 1$ by $- 1$ .

$\arcsin (x) x]_{0}^{1} + 1 \int_{1}^{0} \frac{1}{2 \sqrt{u}} d u$

Step 6.2

Multiply $\int_{1}^{0} \frac{1}{2 \sqrt{u}} d u$ by $1$ .

$\arcsin (x) x]_{0}^{1} + \int_{1}^{0} \frac{1}{2 \sqrt{u}} d u$

Step 7

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

$\arcsin (x) x]_{0}^{1} + \frac{1}{2} \int_{1}^{0} \frac{1}{\sqrt{u}} d u$

Step 8

Apply basic rules of exponents.

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Step 8.1

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

$\arcsin (x) x]_{0}^{1} + \frac{1}{2} \int_{1}^{0} \frac{1}{u^{\frac{1}{2}}} d u$

Step 8.2

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

$\arcsin (x) x]_{0}^{1} + \frac{1}{2} \int_{1}^{0} {(u^{\frac{1}{2}})}^{- 1} d u$

Step 8.3

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

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Step 8.3.1

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

$\arcsin (x) x]_{0}^{1} + \frac{1}{2} \int_{1}^{0} u^{\frac{1}{2} \cdot - 1} d u$

Step 8.3.2

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

$\arcsin (x) x]_{0}^{1} + \frac{1}{2} \int_{1}^{0} u^{\frac{- 1}{2}} d u$

Step 8.3.3

Move the negative in front of the fraction.

$\arcsin (x) x]_{0}^{1} + \frac{1}{2} \int_{1}^{0} u^{- \frac{1}{2}} d u$

Step 9

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

$\arcsin (x) x]_{0}^{1} + \frac{1}{2} 2 u^{\frac{1}{2}}]_{1}^{0}$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\arcsin (x) x$ at $1$ and at $0$ .

$(\arcsin (1) \cdot 1) - \arcsin (0) \cdot 0 + \frac{1}{2} 2 u^{\frac{1}{2}}]_{1}^{0}$

Step 10.2

Evaluate $2 u^{\frac{1}{2}}$ at $0$ and at $1$ .

$(\arcsin (1) \cdot 1) - \arcsin (0) \cdot 0 + \frac{1}{2} ((2 \cdot 0^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3

Simplify.

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Step 10.3.1

Multiply $\arcsin (1)$ by $1$ .

$\arcsin (1) - \arcsin (0) \cdot 0 + \frac{1}{2} ((2 \cdot 0^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.2

Multiply $0$ by $- 1$ .

$\arcsin (1) + 0 \arcsin (0) + \frac{1}{2} ((2 \cdot 0^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.3

Multiply $0$ by $\arcsin (0)$ .

$\arcsin (1) + 0 + \frac{1}{2} ((2 \cdot 0^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.4

Add $\arcsin (1)$ and $0$ .

$\arcsin (1) + \frac{1}{2} ((2 \cdot 0^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.5

Rewrite $0$ as $0^{2}$ .

$\arcsin (1) + \frac{1}{2} (2 \cdot {(0^{2})}^{\frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.6

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

$\arcsin (1) + \frac{1}{2} (2 \cdot 0^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.7

Cancel the common factor of $2$ .

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Step 10.3.7.1

Cancel the common factor.

$\arcsin (1) + \frac{1}{2} (2 \cdot 0^{2 (\frac{1}{2})} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.7.2

Rewrite the expression.

$\arcsin (1) + \frac{1}{2} (2 \cdot 0^{1} - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.8

Evaluate the exponent.

$\arcsin (1) + \frac{1}{2} (2 \cdot 0 - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.9

Multiply $2$ by $0$ .

$\arcsin (1) + \frac{1}{2} (0 - 2 \cdot 1^{\frac{1}{2}})$

Step 10.3.10

One to any power is one.

$\arcsin (1) + \frac{1}{2} (0 - 2 \cdot 1)$

Step 10.3.11

Multiply $- 2$ by $1$ .

$\arcsin (1) + \frac{1}{2} (0 - 2)$

Step 10.3.12

Subtract $2$ from $0$ .

$\arcsin (1) + \frac{1}{2} \cdot - 2$

Step 10.3.13

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

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

Step 10.3.14

Cancel the common factor of $- 2$ and $2$ .

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Step 10.3.14.1

Factor $2$ out of $- 2$ .

$\arcsin (1) + \frac{2 \cdot - 1}{2}$

Step 10.3.14.2

Cancel the common factors.

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Step 10.3.14.2.1

Factor $2$ out of $2$ .

$\arcsin (1) + \frac{2 \cdot - 1}{2 (1)}$

Step 10.3.14.2.2

Cancel the common factor.

$\arcsin (1) + \frac{2 \cdot - 1}{2 \cdot 1}$

Step 10.3.14.2.3

Rewrite the expression.

$\arcsin (1) + \frac{- 1}{1}$

Step 10.3.14.2.4

Divide $- 1$ by $1$ .

$\arcsin (1) - 1$

Step 11

The exact value of $\arcsin (1)$ is $\frac{π}{2}$ .

$\frac{π}{2} - 1$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2} - 1$

Decimal Form:

$0.57079632 \dots$