Calculus Examples

$\int \cos (8 x) d x$ , $u = 8 x$

Step 1

Let $u = 8 x$ . Then $d u = 8 d x$ . Rewrite using $u$ and $d u$ .

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Step 1.1

Let $u = 8 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $8 x$ .

$\frac{d}{d x} [8 x]$

Step 1.1.2

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$8 \frac{d}{d x} [x]$

Step 1.1.3

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

$8 \cdot 1$

Step 1.1.4

Multiply $8$ by $1$ .

$8$

Step 1.2

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

$\int \frac{\cos (u)}{8} d u$

Step 2

Since $\frac{1}{8}$ is constant with respect to $u$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{8} \int \cos (u) d u$

Step 3

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

$\frac{1}{8} (\sin (u) + C)$

Step 4

Simplify.

$\frac{1}{8} \sin (u) + C$

Step 5

Replace all occurrences of $u$ with $8 x$ .

$\frac{1}{8} \sin (8 x) + C$