Calculus Examples

$\cos (\frac{1 x}{3})$

Step 1

Multiply $x$ by $1$ .

$\int \cos (\frac{x}{3}) d x$

Step 2

Let $u = \frac{x}{3}$ . Then $d u = \frac{1}{3} d x$ , so $3 d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \frac{x}{3}$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\frac{x}{3}$ .

$\frac{d}{d x} [\frac{x}{3}]$

Step 2.1.2

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x}{3}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [x]$ .

$\frac{1}{3} \frac{d}{d x} [x]$

Step 2.1.3

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

$\frac{1}{3} \cdot 1$

Step 2.1.4

Multiply $\frac{1}{3}$ by $1$ .

$\frac{1}{3}$

Step 2.2

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

$\int \cos (u) \frac{1}{\frac{1}{3}} d u$

Step 3

Simplify.

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Step 3.1

Multiply by the reciprocal of the fraction to divide by $\frac{1}{3}$ .

$\int \cos (u) (1 \cdot 3) d u$

Step 3.2

Multiply $3$ by $1$ .

$\int \cos (u) \cdot 3 d u$

Step 3.3

Move $3$ to the left of $\cos (u)$ .

$\int 3 \cos (u) d u$

Step 4

Since $3$ is constant with respect to $u$ , move $3$ out of the integral.

$3 \int \cos (u) d u$

Step 5

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

$3 (\sin (u) + C)$

Step 6

Simplify.

$3 \sin (u) + C$

Step 7

Replace all occurrences of $u$ with $\frac{x}{3}$ .

$3 \sin (\frac{x}{3}) + C$

Step 8

Reorder terms.

$3 \sin (\frac{1}{3} x) + C$