Calculus Examples

$f (x) = \frac{x^{3}}{2} - \cos (3 x)$

Step 1

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

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int \frac{x^{3}}{2} - \cos (3 x) d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{x^{3}}{2} d x + \int - \cos (3 x) d x$

Step 4

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int x^{3} d x + \int - \cos (3 x) d x$

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$\frac{1}{2} (\frac{1}{4} x^{4} + C) + \int - \cos (3 x) d x$

Step 6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{2} (\frac{1}{4} x^{4} + C) - \int \cos (3 x) d x$

Step 7

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

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Step 7.1

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

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Step 7.1.1

Differentiate $3 x$ .

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

Step 7.1.2

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

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

Step 7.1.3

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

$3 \cdot 1$

Step 7.1.4

Multiply $3$ by $1$ .

$3$

Step 7.2

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

$\frac{1}{2} (\frac{1}{4} x^{4} + C) - \int \cos (u) \frac{1}{3} d u$

Step 8

Combine $\cos (u)$ and $\frac{1}{3}$ .

$\frac{1}{2} (\frac{1}{4} x^{4} + C) - \int \frac{\cos (u)}{3} d u$

Step 9

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{2} (\frac{1}{4} x^{4} + C) - (\frac{1}{3} \int \cos (u) d u)$

Step 10

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

$\frac{1}{2} (\frac{1}{4} x^{4} + C) - \frac{1}{3} (\sin (u) + C)$

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{1}{2} \cdot \frac{1}{4} x^{4} - \frac{1}{3} \sin (u) + C$

Step 11.2

Simplify.

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Step 11.2.1

Multiply $\frac{1}{2}$ by $\frac{1}{4}$ .

$\frac{1}{2 \cdot 4} x^{4} - \frac{1}{3} \sin (u) + C$

Step 11.2.2

Multiply $2$ by $4$ .

$\frac{1}{8} x^{4} - \frac{1}{3} \sin (u) + C$

Step 12

Replace all occurrences of $u$ with $3 x$ .

$\frac{1}{8} x^{4} - \frac{1}{3} \sin (3 x) + C$

Step 13

The answer is the antiderivative of the function $f (x) = \frac{x^{3}}{2} - \cos (3 x)$ .

$F (x) =$ $\frac{1}{8} x^{4} - \frac{1}{3} \sin (3 x) + C$