Calculus Examples

$\int_{0}^{\frac{π^{2}}{4}} \frac{\sin (\sqrt{x})}{\sqrt{x}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

$\int_{0}^{\frac{π^{2}}{4}} \frac{\sin (x^{\frac{1}{2}})}{\sqrt{x}} d x$

Step 1.2

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

$\int_{0}^{\frac{π^{2}}{4}} \frac{\sin (x^{\frac{1}{2}})}{x^{\frac{1}{2}}} d x$

Step 1.3

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

$\int_{0}^{\frac{π^{2}}{4}} \sin (x^{\frac{1}{2}}) {(x^{\frac{1}{2}})}^{- 1} d x$

Step 1.4

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

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Step 1.4.1

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

$\int_{0}^{\frac{π^{2}}{4}} \sin (x^{\frac{1}{2}}) x^{\frac{1}{2} \cdot - 1} d x$

Step 1.4.2

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

$\int_{0}^{\frac{π^{2}}{4}} \sin (x^{\frac{1}{2}}) x^{\frac{- 1}{2}} d x$

Step 1.4.3

Move the negative in front of the fraction.

$\int_{0}^{\frac{π^{2}}{4}} \sin (x^{\frac{1}{2}}) x^{- \frac{1}{2}} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $x^{\frac{1}{2}}$ .

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{1}{2}$ .

$\frac{1}{2} x^{\frac{1}{2} - 1}$

Step 2.1.3

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

$\frac{1}{2} x^{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 2.1.4

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

$\frac{1}{2} x^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 2.1.5

Combine the numerators over the common denominator.

$\frac{1}{2} x^{\frac{1 - 1 \cdot 2}{2}}$

Step 2.1.6

Simplify the numerator.

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Step 2.1.6.1

Multiply $- 1$ by $2$ .

$\frac{1}{2} x^{\frac{1 - 2}{2}}$

Step 2.1.6.2

Subtract $2$ from $1$ .

$\frac{1}{2} x^{\frac{- 1}{2}}$

Step 2.1.7

Move the negative in front of the fraction.

$\frac{1}{2} x^{- \frac{1}{2}}$

Step 2.1.8

Simplify.

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Step 2.1.8.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{2} \cdot \frac{1}{x^{\frac{1}{2}}}$

Step 2.1.8.2

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

$\frac{1}{2 x^{\frac{1}{2}}}$

Step 2.2

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

$u_{lower} = 0^{\frac{1}{2}}$

Step 2.3

Simplify.

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Step 2.3.1

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

$u_{lower} = {(0^{2})}^{\frac{1}{2}}$

Step 2.3.2

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

$u_{lower} = 0^{2 (\frac{1}{2})}$

Step 2.3.3

Cancel the common factor of $2$ .

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Step 2.3.3.1

Cancel the common factor.

$u_{lower} = 0^{2 (\frac{1}{2})}$

Step 2.3.3.2

Rewrite the expression.

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

Step 2.3.4

Evaluate the exponent.

$u_{lower} = 0$

Step 2.4

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

$u_{upper} = {(\frac{π^{2}}{4})}^{\frac{1}{2}}$

Step 2.5

Simplify.

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Step 2.5.1

Apply the product rule to $\frac{π^{2}}{4}$ .

$u_{upper} = \frac{{(π^{2})}^{\frac{1}{2}}}{4^{\frac{1}{2}}}$

Step 2.5.2

Simplify the numerator.

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Step 2.5.2.1

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

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Step 2.5.2.1.1

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

$u_{upper} = \frac{π^{2 (\frac{1}{2})}}{4^{\frac{1}{2}}}$

Step 2.5.2.1.2

Cancel the common factor of $2$ .

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Step 2.5.2.1.2.1

Cancel the common factor.

$u_{upper} = \frac{π^{2 (\frac{1}{2})}}{4^{\frac{1}{2}}}$

Step 2.5.2.1.2.2

Rewrite the expression.

$u_{upper} = \frac{π^{1}}{4^{\frac{1}{2}}}$

Step 2.5.2.2

Simplify.

$u_{upper} = \frac{π}{4^{\frac{1}{2}}}$

Step 2.5.3

Simplify the denominator.

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Step 2.5.3.1

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

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

Step 2.5.3.2

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

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

Step 2.5.3.3

Cancel the common factor of $2$ .

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Step 2.5.3.3.1

Cancel the common factor.

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

Step 2.5.3.3.2

Rewrite the expression.

$u_{upper} = \frac{π}{2^{1}}$

Step 2.5.3.4

Evaluate the exponent.

$u_{upper} = \frac{π}{2}$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = \frac{π}{2}$

Step 2.7

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

$\int_{0}^{\frac{π}{2}} 2 \sin (u) d u$

Step 3

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

$2 \int_{0}^{\frac{π}{2}} \sin (u) d u$

Step 4

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

$2 (- \cos (u)]_{0}^{\frac{π}{2}})$

Step 5

Evaluate $- \cos (u)$ at $\frac{π}{2}$ and at $0$ .

$2 (- \cos (\frac{π}{2}) + \cos (0))$

Step 6

Simplify.

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Step 6.1

The exact value of $\cos (\frac{π}{2})$ is $0$ .

$2 (- 0 + \cos (0))$

Step 6.2

The exact value of $\cos (0)$ is $1$ .

$2 (- 0 + 1)$

Step 6.3

Multiply $- 1$ by $0$ .

$2 (0 + 1)$

Step 6.4

Add $0$ and $1$ .

$2 \cdot 1$

Step 6.5

Multiply $2$ by $1$ .

$2$