Calculus Examples

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

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\cos (x)$ .

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

Step 1.1.2

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

$- \sin (x)$

Step 1.2

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

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

Step 1.3

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

$u_{lower} = 1$

Step 1.4

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

$u_{upper} = \cos (2 π)$

Step 1.5

Simplify.

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Step 1.5.1

Subtract full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

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

Step 1.5.2

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

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 1$

Step 1.7

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Substitute and simplify.

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Step 5.1

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

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

Step 5.2

Simplify.

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Step 5.2.1

One to any power is one.

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

Step 5.2.2

One to any power is one.

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

Step 5.2.3

Combine the numerators over the common denominator.

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

Step 5.2.4

Subtract $1$ from $1$ .

$- \frac{0}{3}$

Step 5.2.5

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

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Step 5.2.5.1

Factor $3$ out of $0$ .

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

Step 5.2.5.2

Cancel the common factors.

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Step 5.2.5.2.1

Factor $3$ out of $3$ .

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

Step 5.2.5.2.2

Cancel the common factor.

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

Step 5.2.5.2.3

Rewrite the expression.

$- \frac{0}{1}$

Step 5.2.5.2.4

Divide $0$ by $1$ .

$- 0$

Step 5.2.6

Multiply $- 1$ by $0$ .

$0$