Calculus Examples

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

Step 1

Use the half-angle formula to rewrite $\sin^{2} (x)$ as $\frac{1 - \cos (2 x)}{2}$ .

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

Step 2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

$\frac{1}{2} (x]_{0}^{4} + \int_{0}^{4} - \cos (2 x) d x)$

Step 5

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

$\frac{1}{2} (x]_{0}^{4} - \int_{0}^{4} \cos (2 x) d x)$

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $2 x$ .

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

Step 6.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 6.1.3

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

$2 \cdot 1$

Step 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 6.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 6.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \cdot 4$

Step 6.5

Multiply $2$ by $4$ .

$u_{upper} = 8$

Step 6.6

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

$u_{lower} = 0$

$u_{upper} = 8$

Step 6.7

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

$\frac{1}{2} (x]_{0}^{4} - \int_{0}^{8} \cos (u) \frac{1}{2} d u)$

Step 7

Combine $\cos (u)$ and $\frac{1}{2}$ .

$\frac{1}{2} (x]_{0}^{4} - \int_{0}^{8} \frac{\cos (u)}{2} d u)$

Step 8

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} (x]_{0}^{4} - (\frac{1}{2} \int_{0}^{8} \cos (u) d u))$

Step 9

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

$\frac{1}{2} (x]_{0}^{4} - \frac{1}{2} \sin (u)]_{0}^{8})$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $x$ at $4$ and at $0$ .

$\frac{1}{2} ((4) + 0 - \frac{1}{2} \sin (u)]_{0}^{8})$

Step 10.2

Evaluate $\sin (u)$ at $8$ and at $0$ .

$\frac{1}{2} ((4) + 0 - \frac{1}{2} ((\sin (8)) - \sin (0)))$

Step 10.3

Add $4$ and $0$ .

$\frac{1}{2} (4 - \frac{1}{2} (\sin (8) - \sin (0)))$

Step 11

Simplify.

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Step 11.1

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

$\frac{1}{2} (4 - \frac{1}{2} (\sin (8) - 0))$

Step 11.2

Multiply $- 1$ by $0$ .

$\frac{1}{2} (4 - \frac{1}{2} (\sin (8) + 0))$

Step 11.3

Add $\sin (8)$ and $0$ .

$\frac{1}{2} (4 - \frac{1}{2} \sin (8))$

Step 11.4

Combine $\sin (8)$ and $\frac{1}{2}$ .

$\frac{1}{2} (4 - \frac{\sin (8)}{2})$

Step 12

Simplify.

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Step 12.1

Evaluate $\sin (8)$ .

$\frac{1}{2} (4 - \frac{0.98935824}{2})$

Step 12.2

Divide $0.98935824$ by $2$ .

$\frac{1}{2} (4 - 1 \cdot 0.49467912)$

Step 12.3

Multiply $- 1$ by $0.49467912$ .

$\frac{1}{2} (4 - 0.49467912)$

Step 12.4

Subtract $0.49467912$ from $4$ .

$\frac{1}{2} \cdot 3.50532087$

Step 12.5

Combine $\frac{1}{2}$ and $3.50532087$ .

$\frac{3.50532087}{2}$

Step 12.6

Divide $3.50532087$ by $2$ .

$1.75266043$