Calculus Examples

${(6 x + 5)}^{2}$

Step 1

Write ${(6 x + 5)}^{2}$ as a function.

$f (x) = {(6 x + 5)}^{2}$

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.

$F (x) = \int {(6 x + 5)}^{2} d x$

Step 4

Let $u = 6 x + 5$ . Then $d u = 6 d x$ , so $\frac{1}{6} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 4.1

Let $u = 6 x + 5$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 4.1.1

Differentiate $6 x + 5$ .

$\frac{d}{d x} [6 x + 5]$

Step 4.1.2

By the Sum Rule, the derivative of $6 x + 5$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [5]$ .

$\frac{d}{d x} [6 x] + \frac{d}{d x} [5]$

Step 4.1.3

Evaluate $\frac{d}{d x} [6 x]$ .

Tap for more steps...

Step 4.1.3.1

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$6 \frac{d}{d x} [x] + \frac{d}{d x} [5]$

Step 4.1.3.2

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

$6 \cdot 1 + \frac{d}{d x} [5]$

Step 4.1.3.3

Multiply $6$ by $1$ .

$6 + \frac{d}{d x} [5]$

Step 4.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 4.1.4.1

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

$6 + 0$

Step 4.1.4.2

Add $6$ and $0$ .

$6$

Step 4.2

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

$\int u^{2} \frac{1}{6} d u$

Step 5

Combine $u^{2}$ and $\frac{1}{6}$ .

$\int \frac{u^{2}}{6} d u$

Step 6

Since $\frac{1}{6}$ is constant with respect to $u$ , move $\frac{1}{6}$ out of the integral.

$\frac{1}{6} \int u^{2} d u$

Step 7

By the Power Rule, the integral of $u^{2}$ with respect to $u$ is $\frac{1}{3} u^{3}$ .

$\frac{1}{6} (\frac{1}{3} u^{3} + C)$

Step 8

Simplify.

Tap for more steps...

Step 8.1

Rewrite $\frac{1}{6} (\frac{1}{3} u^{3} + C)$ as $\frac{1}{6} \cdot \frac{1}{3} u^{3} + C$ .

$\frac{1}{6} \cdot \frac{1}{3} u^{3} + C$

Step 8.2

Simplify.

Tap for more steps...

Step 8.2.1

Multiply $\frac{1}{6}$ by $\frac{1}{3}$ .

$\frac{1}{6 \cdot 3} u^{3} + C$

Step 8.2.2

Multiply $6$ by $3$ .

$\frac{1}{18} u^{3} + C$

Step 9

Replace all occurrences of $u$ with $6 x + 5$ .

$\frac{1}{18} {(6 x + 5)}^{3} + C$

Step 10

The answer is the antiderivative of the function $f (x) = {(6 x + 5)}^{2}$ .

$F (x) =$ $\frac{1}{18} {(6 x + 5)}^{3} + C$