Calculus Examples

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

Step 1

Factor out $\cos^{2} (x)$ .

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

Step 2

Using the Pythagorean Identity, rewrite $\cos^{2} (x)$ as $1 - \sin^{2} (x)$ .

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

Step 3

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 3.1

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

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Step 3.1.1

Differentiate $\sin (x)$ .

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

Step 3.1.2

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

$\cos (x)$

Step 3.2

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

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

Step 3.3

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

$u_{lower} = 0$

Step 3.4

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

$u_{upper} = \sin (\frac{π}{2})$

Step 3.5

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

$u_{upper} = 1$

Step 3.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 3.7

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

$\int_{0}^{1} 1 - u^{2} d u$

Step 4

Split the single integral into multiple integrals.

$\int_{0}^{1} d u + \int_{0}^{1} - u^{2} d u$

Step 5

Apply the constant rule.

$u]_{0}^{1} + \int_{0}^{1} - u^{2} d u$

Step 6

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

$u]_{0}^{1} - \int_{0}^{1} u^{2} d u$

Step 7

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$u]_{0}^{1} - (\frac{1}{3} u^{3}]_{0}^{1})$

Step 8

Combine $\frac{1}{3}$ and $u^{3}$ .

$u]_{0}^{1} - (\frac{u^{3}}{3}]_{0}^{1})$

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $u$ at $1$ and at $0$ .

$(1) + 0 - (\frac{u^{3}}{3}]_{0}^{1})$

Step 9.2

Evaluate $\frac{u^{3}}{3}$ at $1$ and at $0$ .

$1 + 0 - (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 9.3

Simplify.

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Step 9.3.1

Add $1$ and $0$ .

$1 - (\frac{1^{3}}{3} - \frac{0^{3}}{3})$

Step 9.3.2

One to any power is one.

$1 - (\frac{1}{3} - \frac{0^{3}}{3})$

Step 9.3.3

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

$1 - (\frac{1}{3} - \frac{0}{3})$

Step 9.3.4

Cancel the common factor of $0$ and $3$ .

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Step 9.3.4.1

Factor $3$ out of $0$ .

$1 - (\frac{1}{3} - \frac{3 (0)}{3})$

Step 9.3.4.2

Cancel the common factors.

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Step 9.3.4.2.1

Factor $3$ out of $3$ .

$1 - (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 9.3.4.2.2

Cancel the common factor.

$1 - (\frac{1}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 9.3.4.2.3

Rewrite the expression.

$1 - (\frac{1}{3} - \frac{0}{1})$

Step 9.3.4.2.4

Divide $0$ by $1$ .

$1 - (\frac{1}{3} - 0)$

Step 9.3.5

Multiply $- 1$ by $0$ .

$1 - (\frac{1}{3} + 0)$

Step 9.3.6

Add $\frac{1}{3}$ and $0$ .

$1 - \frac{1}{3}$

Step 9.3.7

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

$\frac{3}{3} - \frac{1}{3}$

Step 9.3.8

Combine the numerators over the common denominator.

$\frac{3 - 1}{3}$

Step 9.3.9

Subtract $1$ from $3$ .

$\frac{2}{3}$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$