Calculus Examples

$g (x) = - 4 \cos (4 x)$

Step 1

The function $G (x)$ can be found by finding the indefinite integral of the derivative $g (x)$ .

$G (x) = \int g (x) d x$

Step 2

Set up the integral to solve.

$G (x) = \int - 4 \cos (4 x) d x$

Step 3

Since $- 4$ is constant with respect to $x$ , move $- 4$ out of the integral.

$- 4 \int \cos (4 x) d x$

Step 4

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 4.1

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

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Step 4.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 4.1.2

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 4.1.3

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 4.1.4

Multiply $4$ by $1$ .

$4$

Step 4.2

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

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

Step 5

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

$- 4 \int \frac{\cos (u)}{4} d u$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Combine $\frac{1}{4}$ and $- 4$ .

$\frac{- 4}{4} \int \cos (u) d u$

Step 7.2

Cancel the common factor of $- 4$ and $4$ .

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Step 7.2.1

Factor $4$ out of $- 4$ .

$\frac{4 \cdot - 1}{4} \int \cos (u) d u$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot - 1}{4 (1)} \int \cos (u) d u$

Step 7.2.2.2

Cancel the common factor.

$\frac{4 \cdot - 1}{4 \cdot 1} \int \cos (u) d u$

Step 7.2.2.3

Rewrite the expression.

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

Step 7.2.2.4

Divide $- 1$ by $1$ .

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

Step 8

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

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

Step 9

Simplify.

$- \sin (u) + C$

Step 10

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

$- \sin (4 x) + C$

Step 11

The answer is the antiderivative of the function $g (x) = - 4 \cos (4 x)$ .

$G (x) =$ $- \sin (4 x) + C$