Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

$- \cos (x)]_{0}^{\frac{π}{2}} + \int_{0}^{\frac{π}{2}} \cos (x) \sin (x) d x$

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $\cos (x)$ .

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

Step 3.1.2

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

$- \sin (x)$

Step 3.2

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

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

Step 3.3

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

$u_{lower} = 1$

Step 3.4

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

$u_{upper} = \cos (\frac{π}{2})$

Step 3.5

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

$u_{upper} = 0$

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.

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

Step 4

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

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

Step 5

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

$- \cos (x)]_{0}^{\frac{π}{2}} - (\frac{1}{2} u^{2}]_{1}^{0})$

Step 6

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

$- \cos (x)]_{0}^{\frac{π}{2}} - (\frac{u^{2}}{2}]_{1}^{0})$

Step 7

Substitute and simplify.

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Step 7.1

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

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

Step 7.2

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

$(- \cos (\frac{π}{2})) + \cos (0) - ((\frac{0^{2}}{2}) - \frac{1^{2}}{2})$

Step 7.3

Simplify.

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Step 7.3.1

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

$- \cos (\frac{π}{2}) + \cos (0) - (\frac{0}{2} - \frac{1^{2}}{2})$

Step 7.3.2

Cancel the common factor of $0$ and $2$ .

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Step 7.3.2.1

Factor $2$ out of $0$ .

$- \cos (\frac{π}{2}) + \cos (0) - (\frac{2 (0)}{2} - \frac{1^{2}}{2})$

Step 7.3.2.2

Cancel the common factors.

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Step 7.3.2.2.1

Factor $2$ out of $2$ .

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

Step 7.3.2.2.2

Cancel the common factor.

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

Step 7.3.2.2.3

Rewrite the expression.

$- \cos (\frac{π}{2}) + \cos (0) - (\frac{0}{1} - \frac{1^{2}}{2})$

Step 7.3.2.2.4

Divide $0$ by $1$ .

$- \cos (\frac{π}{2}) + \cos (0) - (0 - \frac{1^{2}}{2})$

Step 7.3.3

One to any power is one.

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

Step 7.3.4

Subtract $\frac{1}{2}$ from $0$ .

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

Step 7.3.5

Multiply $- 1$ by $- 1$ .

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

Step 7.3.6

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

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

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

$- 0 + 1 + \frac{1}{2}$

Step 8.3

Multiply $- 1$ by $0$ .

$0 + 1 + \frac{1}{2}$

Step 8.4

Add $0$ and $1$ .

$1 + \frac{1}{2}$

Step 8.5

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

$\frac{2}{2} + \frac{1}{2}$

Step 8.6

Combine the numerators over the common denominator.

$\frac{2 + 1}{2}$

Step 8.7

Add $2$ and $1$ .

$\frac{3}{2}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{3}{2}$

Decimal Form:

$1.5$

Mixed Number Form:

$1 \frac{1}{2}$