Calculus Examples

$\int_{0}^{\frac{π}{2}} 5 {\sin (x)}^{\frac{3}{2}} \cos (x) d x$

Step 1

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{0}^{\frac{π}{2}} {\sin (x)}^{\frac{3}{2}} \cos (x) d x$

Step 2

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 2.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 2.2

Substitute the lower limit in for $x$ in $u = \sin (x)$ .

$u_{lower} = \sin (0)$

Step 2.3

The exact value of $\sin (0)$ is $0$ .

$u_{lower} = 0$

Step 2.4

Substitute the upper limit in for $x$ in $u = \sin (x)$ .

$u_{upper} = \sin (\frac{π}{2})$

Step 2.5

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

$u_{upper} = 1$

Step 2.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 2.7

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

$5 \int_{0}^{1} u^{\frac{3}{2}} d u$

Step 3

By the Power Rule, the integral of $u^{\frac{3}{2}}$ with respect to $u$ is $\frac{2}{5} u^{\frac{5}{2}}$ .

$5 (\frac{2}{5} u^{\frac{5}{2}}]_{0}^{1})$

Step 4

Combine $\frac{2}{5}$ and $u^{\frac{5}{2}}$ .

$5 (\frac{2 u^{\frac{5}{2}}}{5}]_{0}^{1})$

Step 5

Substitute and simplify.

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Step 5.1

Evaluate $\frac{2 u^{\frac{5}{2}}}{5}$ at $1$ and at $0$ .

$5 ((\frac{2 \cdot 1^{\frac{5}{2}}}{5}) - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 5.2

Simplify.

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Step 5.2.1

One to any power is one.

$5 (\frac{2 \cdot 1}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 5.2.2

Multiply $2$ by $1$ .

$5 (\frac{2}{5} - \frac{2 \cdot 0^{\frac{5}{2}}}{5})$

Step 5.2.3

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

$5 (\frac{2}{5} - \frac{2 \cdot {(0^{2})}^{\frac{5}{2}}}{5})$

Step 5.2.4

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

$5 (\frac{2}{5} - \frac{2 \cdot 0^{2 (\frac{5}{2})}}{5})$

Step 5.2.5

Cancel the common factor of $2$ .

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Step 5.2.5.1

Cancel the common factor.

$5 (\frac{2}{5} - \frac{2 \cdot 0^{2 (\frac{5}{2})}}{5})$

Step 5.2.5.2

Rewrite the expression.

$5 (\frac{2}{5} - \frac{2 \cdot 0^{5}}{5})$

Step 5.2.6

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

$5 (\frac{2}{5} - \frac{2 \cdot 0}{5})$

Step 5.2.7

Multiply $2$ by $0$ .

$5 (\frac{2}{5} - \frac{0}{5})$

Step 5.2.8

Cancel the common factor of $0$ and $5$ .

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Step 5.2.8.1

Factor $5$ out of $0$ .

$5 (\frac{2}{5} - \frac{5 (0)}{5})$

Step 5.2.8.2

Cancel the common factors.

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Step 5.2.8.2.1

Factor $5$ out of $5$ .

$5 (\frac{2}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 5.2.8.2.2

Cancel the common factor.

$5 (\frac{2}{5} - \frac{5 \cdot 0}{5 \cdot 1})$

Step 5.2.8.2.3

Rewrite the expression.

$5 (\frac{2}{5} - \frac{0}{1})$

Step 5.2.8.2.4

Divide $0$ by $1$ .

$5 (\frac{2}{5} - 0)$

Step 5.2.9

Multiply $- 1$ by $0$ .

$5 (\frac{2}{5} + 0)$

Step 5.2.10

Add $\frac{2}{5}$ and $0$ .

$5 (\frac{2}{5})$

Step 5.2.11

Combine $5$ and $\frac{2}{5}$ .

$\frac{5 \cdot 2}{5}$

Step 5.2.12

Multiply $5$ by $2$ .

$\frac{10}{5}$

Step 5.2.13

Cancel the common factor of $10$ and $5$ .

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Step 5.2.13.1

Factor $5$ out of $10$ .

$\frac{5 \cdot 2}{5}$

Step 5.2.13.2

Cancel the common factors.

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Step 5.2.13.2.1

Factor $5$ out of $5$ .

$\frac{5 \cdot 2}{5 (1)}$

Step 5.2.13.2.2

Cancel the common factor.

$\frac{5 \cdot 2}{5 \cdot 1}$

Step 5.2.13.2.3

Rewrite the expression.

$\frac{2}{1}$

Step 5.2.13.2.4

Divide $2$ by $1$ .

$2$