Calculus Examples

$\int_{0}^{π} \cos^{2} (x) d x$

Step 1

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

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

Step 2

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

Apply the constant rule.

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

Step 5

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

Tap for more steps...

Step 5.1

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

Tap for more steps...

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 $u = 2 x$ .

$u_{upper} = 2 π$

Step 5.5

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

$u_{lower} = 0$

$u_{upper} = 2 π$

Step 5.6

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Substitute and simplify.

Tap for more steps...

Step 9.1

Evaluate $x$ at $π$ and at $0$ .

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

Step 9.2

Evaluate $\sin (u)$ at $2 π$ and at $0$ .

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

Step 9.3

Add $π$ and $0$ .

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

Step 10

Simplify.

Tap for more steps...

Step 10.1

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

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

Step 10.2

Multiply $- 1$ by $0$ .

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

Step 10.3

Add $\sin (2 π)$ and $0$ .

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

Step 10.4

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

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

Step 11

Simplify.

Tap for more steps...

Step 11.1

Simplify each term.

Tap for more steps...

Step 11.1.1

Simplify the numerator.

Tap for more steps...

Step 11.1.1.1

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

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

Step 11.1.1.2

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

$\frac{1}{2} (π + \frac{0}{2})$

Step 11.1.2

Divide $0$ by $2$ .

$\frac{1}{2} (π + 0)$

Step 11.2

Add $π$ and $0$ .

$\frac{1}{2} π$

Step 11.3

Combine $\frac{1}{2}$ and $π$ .

$\frac{π}{2}$

Step 12

The result can be shown in multiple forms.

Exact Form:

$\frac{π}{2}$

Decimal Form:

$1.57079632 \dots$