Calculus Examples

$\int_{0}^{2 π} \cos^{2} (x) d x$

Step 1

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

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

Step 2

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

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

Step 5

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 5.1

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

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Step 5.1.1

Differentiate $2 x$ .

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

Step 5.1.2

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 5.1.3

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 5.1.4

Multiply $2$ by $1$ .

$2$

Step 5.2

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

$u_{lower} = 2 \cdot 0$

Step 5.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 5.4

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

$u_{upper} = 2 (2 π)$

Step 5.5

Multiply $2$ by $2$ .

$u_{upper} = 4 π$

Step 5.6

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

$u_{lower} = 0$

$u_{upper} = 4 π$

Step 5.7

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Substitute and simplify.

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Step 9.1

Evaluate $x$ at $2 π$ and at $0$ .

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

Step 9.2

Evaluate $\sin (u)$ at $4 π$ and at $0$ .

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

Step 9.3

Add $2 π$ and $0$ .

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

Step 10

Simplify.

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Step 10.1

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

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

Step 10.2

Multiply $- 1$ by $0$ .

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

Step 10.3

Add $\sin (4 π)$ and $0$ .

$\frac{1}{2} (2 π + \frac{1}{2} \sin (4 π))$

Step 10.4

Combine $\frac{1}{2}$ and $\sin (4 π)$ .

$\frac{1}{2} (2 π + \frac{\sin (4 π)}{2})$

Step 11

Simplify.

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Step 11.1

Simplify each term.

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Step 11.1.1

Simplify the numerator.

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Step 11.1.1.1

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

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

Step 11.1.1.2

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

$\frac{1}{2} (2 π + \frac{0}{2})$

Step 11.1.2

Divide $0$ by $2$ .

$\frac{1}{2} (2 π + 0)$

Step 11.2

Add $2 π$ and $0$ .

$\frac{1}{2} (2 π)$

Step 11.3

Cancel the common factor of $2$ .

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Step 11.3.1

Factor $2$ out of $2 π$ .

$\frac{1}{2} (2 (π))$

Step 11.3.2

Cancel the common factor.

$\frac{1}{2} (2 π)$

Step 11.3.3

Rewrite the expression.

$π$

Step 12

The result can be shown in multiple forms.

Exact Form:

$π$

Decimal Form:

$3.14159265 \dots$