Calculus Examples

$\int_{0}^{\frac{1}{2}} \arccos (x) d x$

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Multiply $- 1$ by $- 1$ .

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

Step 4.2

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

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

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

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

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

Step 5.3

Simplify.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

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

$u_{lower} = 1 - 0$

Step 5.3.1.2

Multiply $- 1$ by $0$ .

$u_{lower} = 1 + 0$

Step 5.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 5.4

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

$u_{upper} = 1 - {(\frac{1}{2})}^{2}$

Step 5.5

Simplify.

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Step 5.5.1

Simplify each term.

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Step 5.5.1.1

Apply the product rule to $\frac{1}{2}$ .

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

Step 5.5.1.2

One to any power is one.

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

Step 5.5.1.3

Raise $2$ to the power of $2$ .

$u_{upper} = 1 - \frac{1}{4}$

Step 5.5.2

Write $1$ as a fraction with a common denominator.

$u_{upper} = \frac{4}{4} - \frac{1}{4}$

Step 5.5.3

Combine the numerators over the common denominator.

$u_{upper} = \frac{4 - 1}{4}$

Step 5.5.4

Subtract $1$ from $4$ .

$u_{upper} = \frac{3}{4}$

Step 5.6

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

$u_{lower} = 1$

$u_{upper} = \frac{3}{4}$

Step 5.7

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

$\arccos (x) x]_{0}^{\frac{1}{2}} + \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 6

Simplify.

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Step 6.1

Move the negative in front of the fraction.

$\arccos (x) x]_{0}^{\frac{1}{2}} + \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} (- \frac{1}{2}) d u$

Step 6.2

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

$\arccos (x) x]_{0}^{\frac{1}{2}} + \int_{1}^{\frac{3}{4}} - \frac{1}{\sqrt{u} \cdot 2} d u$

Step 6.3

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

$\arccos (x) x]_{0}^{\frac{1}{2}} + \int_{1}^{\frac{3}{4}} - \frac{1}{2 \sqrt{u}} d u$

Step 7

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

$\arccos (x) x]_{0}^{\frac{1}{2}} - \int_{1}^{\frac{3}{4}} \frac{1}{2 \sqrt{u}} d u$

Step 8

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

$\arccos (x) x]_{0}^{\frac{1}{2}} - (\frac{1}{2} \int_{1}^{\frac{3}{4}} \frac{1}{\sqrt{u}} d u)$

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 $u^{\frac{1}{2}}$ .

$\arccos (x) x]_{0}^{\frac{1}{2}} - \frac{1}{2} \int_{1}^{\frac{3}{4}} \frac{1}{u^{\frac{1}{2}}} d u$

Step 9.2

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

$\arccos (x) x]_{0}^{\frac{1}{2}} - \frac{1}{2} \int_{1}^{\frac{3}{4}} {(u^{\frac{1}{2}})}^{- 1} d u$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\arccos (x) x]_{0}^{\frac{1}{2}} - \frac{1}{2} \int_{1}^{\frac{3}{4}} u^{\frac{1}{2} \cdot - 1} d u$

Step 9.3.2

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

$\arccos (x) x]_{0}^{\frac{1}{2}} - \frac{1}{2} \int_{1}^{\frac{3}{4}} u^{\frac{- 1}{2}} d u$

Step 9.3.3

Move the negative in front of the fraction.

$\arccos (x) x]_{0}^{\frac{1}{2}} - \frac{1}{2} \int_{1}^{\frac{3}{4}} u^{- \frac{1}{2}} d u$

Step 10

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

$\arccos (x) x]_{0}^{\frac{1}{2}} - \frac{1}{2} 2 u^{\frac{1}{2}}]_{1}^{\frac{3}{4}}$

Step 11

Substitute and simplify.

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Step 11.1

Evaluate $\arccos (x) x$ at $\frac{1}{2}$ and at $0$ .

$(\arccos (\frac{1}{2}) \frac{1}{2}) - \arccos (0) \cdot 0 - \frac{1}{2} 2 u^{\frac{1}{2}}]_{1}^{\frac{3}{4}}$

Step 11.2

Evaluate $2 u^{\frac{1}{2}}$ at $\frac{3}{4}$ and at $1$ .

$(\arccos (\frac{1}{2}) \frac{1}{2}) - \arccos (0) \cdot 0 - \frac{1}{2} ((2 {(\frac{3}{4})}^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 11.3

Simplify.

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Step 11.3.1

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

$\frac{\arccos (\frac{1}{2})}{2} - \arccos (0) \cdot 0 - \frac{1}{2} ((2 {(\frac{3}{4})}^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 11.3.2

Multiply $0$ by $- 1$ .

$\frac{\arccos (\frac{1}{2})}{2} + 0 \arccos (0) - \frac{1}{2} ((2 {(\frac{3}{4})}^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 11.3.3

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

$\frac{\arccos (\frac{1}{2})}{2} + 0 - \frac{1}{2} ((2 {(\frac{3}{4})}^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 11.3.4

Add $\frac{\arccos (\frac{1}{2})}{2}$ and $0$ .

$\frac{\arccos (\frac{1}{2})}{2} - \frac{1}{2} ((2 {(\frac{3}{4})}^{\frac{1}{2}}) - 2 \cdot 1^{\frac{1}{2}})$

Step 11.3.5

One to any power is one.

$\frac{\arccos (\frac{1}{2})}{2} - \frac{1}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2 \cdot 1)$

Step 11.3.6

Multiply $- 2$ by $1$ .

$\frac{\arccos (\frac{1}{2})}{2} - \frac{1}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)$

Step 11.3.7

To write $- \frac{1}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\arccos (\frac{1}{2})}{2} - \frac{1}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2) \cdot \frac{2}{2}$

Step 11.3.8

Combine $- \frac{1}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)$ and $\frac{2}{2}$ .

$\frac{\arccos (\frac{1}{2})}{2} + \frac{- \frac{1}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2) \cdot 2}{2}$

Step 11.3.9

Combine the numerators over the common denominator.

$\frac{\arccos (\frac{1}{2}) - \frac{1}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2) \cdot 2}{2}$

Step 11.3.10

Multiply $2$ by $- 1$ .

$\frac{\arccos (\frac{1}{2}) - 2 (\frac{1}{2}) (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)}{2}$

Step 11.3.11

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

$\frac{\arccos (\frac{1}{2}) + \frac{- 2}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)}{2}$

Step 11.3.12

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

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Step 11.3.12.1

Factor $2$ out of $- 2$ .

$\frac{\arccos (\frac{1}{2}) + \frac{2 \cdot - 1}{2} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)}{2}$

Step 11.3.12.2

Cancel the common factors.

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Step 11.3.12.2.1

Factor $2$ out of $2$ .

$\frac{\arccos (\frac{1}{2}) + \frac{2 \cdot - 1}{2 (1)} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)}{2}$

Step 11.3.12.2.2

Cancel the common factor.

$\frac{\arccos (\frac{1}{2}) + \frac{2 \cdot - 1}{2 \cdot 1} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)}{2}$

Step 11.3.12.2.3

Rewrite the expression.

$\frac{\arccos (\frac{1}{2}) + \frac{- 1}{1} (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)}{2}$

Step 11.3.12.2.4

Divide $- 1$ by $1$ .

$\frac{\arccos (\frac{1}{2}) - (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)}{2}$

Step 12

Simplify.

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Step 12.1

Simplify the numerator.

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Step 12.1.1

The exact value of $\arccos (\frac{1}{2})$ is $\frac{π}{3}$ .

$\frac{\frac{π}{3} - (2 {(\frac{3}{4})}^{\frac{1}{2}} - 2)}{2}$

Step 12.1.2

Simplify each term.

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Step 12.1.2.1

Apply the product rule to $\frac{3}{4}$ .

$\frac{\frac{π}{3} - (2 \frac{3^{\frac{1}{2}}}{4^{\frac{1}{2}}} - 2)}{2}$

Step 12.1.2.2

Simplify the denominator.

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Step 12.1.2.2.1

Rewrite $4$ as $2^{2}$ .

$\frac{\frac{π}{3} - (2 \frac{3^{\frac{1}{2}}}{{(2^{2})}^{\frac{1}{2}}} - 2)}{2}$

Step 12.1.2.2.2

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

$\frac{\frac{π}{3} - (2 \frac{3^{\frac{1}{2}}}{2^{2 (\frac{1}{2})}} - 2)}{2}$

Step 12.1.2.2.3

Cancel the common factor of $2$ .

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Step 12.1.2.2.3.1

Cancel the common factor.

$\frac{\frac{π}{3} - (2 \frac{3^{\frac{1}{2}}}{2^{2 (\frac{1}{2})}} - 2)}{2}$

Step 12.1.2.2.3.2

Rewrite the expression.

$\frac{\frac{π}{3} - (2 \frac{3^{\frac{1}{2}}}{2^{1}} - 2)}{2}$

Step 12.1.2.2.4

Evaluate the exponent.

$\frac{\frac{π}{3} - (2 \frac{3^{\frac{1}{2}}}{2} - 2)}{2}$

Step 12.1.2.3

Cancel the common factor of $2$ .

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Step 12.1.2.3.1

Cancel the common factor.

$\frac{\frac{π}{3} - (2 \frac{3^{\frac{1}{2}}}{2} - 2)}{2}$

Step 12.1.2.3.2

Rewrite the expression.

$\frac{\frac{π}{3} - (3^{\frac{1}{2}} - 2)}{2}$

Step 12.1.3

Apply the distributive property.

$\frac{\frac{π}{3} - 3^{\frac{1}{2}} - - 2}{2}$

Step 12.1.4

Multiply $- 1$ by $- 2$ .

$\frac{\frac{π}{3} - 3^{\frac{1}{2}} + 2}{2}$

Step 12.1.5

To write $- 3^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{π}{3} - 3^{\frac{1}{2}} \cdot \frac{3}{3} + 2}{2}$

Step 12.1.6

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

$\frac{\frac{π}{3} + \frac{- 3^{\frac{1}{2}} \cdot 3}{3} + 2}{2}$

Step 12.1.7

Combine the numerators over the common denominator.

$\frac{\frac{π - 3^{\frac{1}{2}} \cdot 3}{3} + 2}{2}$

Step 12.1.8

Multiply $3^{\frac{1}{2}}$ by $3$ by adding the exponents.

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Step 12.1.8.1

Move $3$ .

$\frac{\frac{π - (3 \cdot 3^{\frac{1}{2}})}{3} + 2}{2}$

Step 12.1.8.2

Multiply $3$ by $3^{\frac{1}{2}}$ .

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Step 12.1.8.2.1

Raise $3$ to the power of $1$ .

$\frac{\frac{π - (3^{1} \cdot 3^{\frac{1}{2}})}{3} + 2}{2}$

Step 12.1.8.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{π - 3^{1 + \frac{1}{2}}}{3} + 2}{2}$

Step 12.1.8.3

Write $1$ as a fraction with a common denominator.

$\frac{\frac{π - 3^{\frac{2}{2} + \frac{1}{2}}}{3} + 2}{2}$

Step 12.1.8.4

Combine the numerators over the common denominator.

$\frac{\frac{π - 3^{\frac{2 + 1}{2}}}{3} + 2}{2}$

Step 12.1.8.5

Add $2$ and $1$ .

$\frac{\frac{π - 3^{\frac{3}{2}}}{3} + 2}{2}$

Step 12.1.9

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{\frac{π - 3^{\frac{3}{2}}}{3} + 2 \cdot \frac{3}{3}}{2}$

Step 12.1.10

Combine $2$ and $\frac{3}{3}$ .

$\frac{\frac{π - 3^{\frac{3}{2}}}{3} + \frac{2 \cdot 3}{3}}{2}$

Step 12.1.11

Combine the numerators over the common denominator.

$\frac{\frac{π - 3^{\frac{3}{2}} + 2 \cdot 3}{3}}{2}$

Step 12.1.12

Multiply $2$ by $3$ .

$\frac{\frac{π - 3^{\frac{3}{2}} + 6}{3}}{2}$

Step 12.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{π - 3^{\frac{3}{2}} + 6}{3} \cdot \frac{1}{2}$

Step 12.3

Multiply $\frac{π - 3^{\frac{3}{2}} + 6}{3} \cdot \frac{1}{2}$ .

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Step 12.3.1

Multiply $\frac{π - 3^{\frac{3}{2}} + 6}{3}$ by $\frac{1}{2}$ .

$\frac{π - 3^{\frac{3}{2}} + 6}{3 \cdot 2}$

Step 12.3.2

Multiply $3$ by $2$ .

$\frac{π - 3^{\frac{3}{2}} + 6}{6}$

Step 13

The result can be shown in multiple forms.

Exact Form:

$\frac{π - 3^{\frac{3}{2}} + 6}{6}$

Decimal Form:

$0.65757337 \dots$