Calculus Examples

$\int_{0}^{5} \cos (5 x) d x$

Step 1

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 1.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 1.1.3

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

$5 \cdot 1$

Step 1.1.4

Multiply $5$ by $1$ .

$5$

Step 1.2

Substitute the lower limit in for $x$ in $u = 5 x$ .

$u_{lower} = 5 \cdot 0$

Step 1.3

Multiply $5$ by $0$ .

$u_{lower} = 0$

Step 1.4

Substitute the upper limit in for $x$ in $u = 5 x$ .

$u_{upper} = 5 \cdot 5$

Step 1.5

Multiply $5$ by $5$ .

$u_{upper} = 25$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 25$

Step 1.7

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

$\int_{0}^{25} \cos (u) \frac{1}{5} d u$

Step 2

Combine $\cos (u)$ and $\frac{1}{5}$ .

$\int_{0}^{25} \frac{\cos (u)}{5} d u$

Step 3

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{5} \int_{0}^{25} \cos (u) d u$

Step 4

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

$\frac{1}{5} \sin (u)]_{0}^{25}$

Step 5

Evaluate $\sin (u)$ at $25$ and at $0$ .

$\frac{1}{5} (\sin (25) - \sin (0))$

Step 6

Simplify.

Tap for more steps...

Step 6.1

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

$\frac{1}{5} (\sin (25) - 0)$

Step 6.2

Multiply $- 1$ by $0$ .

$\frac{1}{5} (\sin (25) + 0)$

Step 6.3

Add $\sin (25)$ and $0$ .

$\frac{1}{5} \sin (25)$

Step 6.4

Combine $\frac{1}{5}$ and $\sin (25)$ .

$\frac{\sin (25)}{5}$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Evaluate $\sin (25)$ .

$\frac{- 0.13235175}{5}$

Step 7.2

Divide $- 0.13235175$ by $5$ .

$- 0.02647035$