Calculus Examples

$\int \sin (4 x) d x$

Step 1

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

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Step 1.1

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

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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

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

$\int \sin (u) \frac{1}{4} d u$

Step 2

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

$\int \frac{\sin (u)}{4} d u$

Step 3

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

$\frac{1}{4} \int \sin (u) d u$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Simplify.

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

Step 5.2

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

$- \frac{\cos (u)}{4} + C$

Step 6

Replace all occurrences of $u$ with $4 x$ .

$- \frac{\cos (4 x)}{4} + C$

Step 7

Reorder terms.

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

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