Calculus Examples

$\int \cos (\frac{x}{5}) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $\frac{x}{5}$ .

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

Step 1.1.2

Since $\frac{1}{5}$ is constant with respect to $x$ , the derivative of $\frac{x}{5}$ with respect to $x$ is $\frac{1}{5} \frac{d}{d x} [x]$ .

$\frac{1}{5} \frac{d}{d x} [x]$

Step 1.1.3

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

$\frac{1}{5} \cdot 1$

Step 1.1.4

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{5}$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \cos (u) \frac{1}{\frac{1}{5}} d u$

Step 2

Simplify.

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Step 2.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{5}$ .

$\int \cos (u) (1 \cdot 5) d u$

Step 2.2

Multiply $5$ by $1$ .

$\int \cos (u) \cdot 5 d u$

Step 2.3

Move $5$ to the left of $\cos (u)$ .

$\int 5 \cos (u) d u$

Step 3

Since $5$ is constant with respect to $u$ , move $5$ out of the integral.

$5 \int \cos (u) d u$

Step 4

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

$5 (\sin (u) + C)$

Step 5

Simplify.

$5 \sin (u) + C$

Step 6

Replace all occurrences of $u$ with $\frac{x}{5}$ .

$5 \sin (\frac{x}{5}) + C$

Step 7

Reorder terms.

$5 \sin (\frac{1}{5} x) + C$