Calculus Examples

$x (x^{2} - 5)$

Step 1

Write $x (x^{2} - 5)$ as a function.

$f (x) = x (x^{2} - 5)$

Step 2

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 3

Set up the integral to solve.

$F (x) = \int x (x^{2} - 5) d x$

Step 4

Let $u = x^{2} - 5$ . Then $d u = 2 x d x$ , so $\frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = x^{2} - 5$ . Find $\frac{d u}{d x}$ .

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Step 4.1.1

Differentiate $x^{2} - 5$ .

$\frac{d}{d x} [x^{2} - 5]$

Step 4.1.2

By the Sum Rule, the derivative of $x^{2} - 5$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 5]$

Step 4.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x + \frac{d}{d x} [- 5]$

Step 4.1.4

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$2 x + 0$

Step 4.1.5

Add $2 x$ and $0$ .

$2 x$

Step 4.2

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

$\int u \frac{1}{2} d u$

Step 5

Combine $u$ and $\frac{1}{2}$ .

$\int \frac{u}{2} d u$

Step 6

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

$\frac{1}{2} \int u d u$

Step 7

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$\frac{1}{2} (\frac{1}{2} u^{2} + C)$

Step 8

Simplify.

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Step 8.1

Rewrite $\frac{1}{2} (\frac{1}{2} u^{2} + C)$ as $\frac{1}{2} \cdot \frac{1}{2} u^{2} + C$ .

$\frac{1}{2} \cdot \frac{1}{2} u^{2} + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$\frac{1}{2 \cdot 2} u^{2} + C$

Step 8.2.2

Multiply $2$ by $2$ .

$\frac{1}{4} u^{2} + C$

Step 9

Replace all occurrences of $u$ with $x^{2} - 5$ .

$\frac{1}{4} {(x^{2} - 5)}^{2} + C$

Step 10

The answer is the antiderivative of the function $f (x) = x (x^{2} - 5)$ .

$F (x) =$ $\frac{1}{4} {(x^{2} - 5)}^{2} + C$