Calculus Examples

$\int_{0}^{π} \sin (4 x) d x$

Step 1

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 1.1.3

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

$4 \cdot 1$

Step 1.1.4

Multiply $4$ by $1$ .

$4$

Step 1.2

Substitute the lower limit in for $x$ in $u = 4 x$ .

$u_{lower} = 4 \cdot 0$

Step 1.3

Multiply $4$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = 4 x$ .

$u_{upper} = 4 π$

Step 1.5

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 1.6

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

$\int_{0}^{4 π} \sin (u) \frac{1}{4} d u$

Step 2

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

$\int_{0}^{4 π} \frac{\sin (u)}{4} d u$

Step 3

Since $\frac{1}{4}$ is constant with respect to $u$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{0}^{4 π} \sin (u) d u$

Step 4

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

$\frac{1}{4} - \cos (u)]_{0}^{4 π}$

Step 5

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

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

Step 6

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

$\frac{1}{4} (- \cos (4 π) + 1)$

Step 7

Simplify.

Tap for more steps...

Step 7.1

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

$\frac{1}{4} (- \cos (0) + 1)$

Step 7.2

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

$\frac{1}{4} (- 1 \cdot 1 + 1)$

Step 7.3

Multiply $- 1$ by $1$ .

$\frac{1}{4} (- 1 + 1)$

Step 7.4

Add $- 1$ and $1$ .

$\frac{1}{4} \cdot 0$

Step 7.5

Multiply $\frac{1}{4}$ by $0$ .

$0$