Calculus Examples

$\frac{6 x}{{(5 + 3 x^{2})}^{4}}$

Step 1

Write $\frac{6 x}{{(5 + 3 x^{2})}^{4}}$ as a function.

$f (x) = \frac{6 x}{{(5 + 3 x^{2})}^{4}}$

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.

$F (x) = \int \frac{6 x}{{(5 + 3 x^{2})}^{4}} d x$

Step 4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int \frac{x}{{(5 + 3 x^{2})}^{4}} d x$

Step 5

Let $u = 5 + 3 x^{2}$ . Then $d u = 6 x d x$ , so $\frac{1}{6} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.1

Let $u = 5 + 3 x^{2}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 5.1.1

Differentiate $5 + 3 x^{2}$ .

$\frac{d}{d x} [5 + 3 x^{2}]$

Step 5.1.2

Differentiate.

Tap for more steps...

Step 5.1.2.1

By the Sum Rule, the derivative of $5 + 3 x^{2}$ with respect to $x$ is $\frac{d}{d x} [5] + \frac{d}{d x} [3 x^{2}]$ .

$\frac{d}{d x} [5] + \frac{d}{d x} [3 x^{2}]$

Step 5.1.2.2

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

$0 + \frac{d}{d x} [3 x^{2}]$

Step 5.1.3

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

Tap for more steps...

Step 5.1.3.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x^{2}$ with respect to $x$ is $3 \frac{d}{d x} [x^{2}]$ .

$0 + 3 \frac{d}{d x} [x^{2}]$

Step 5.1.3.2

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

$0 + 3 (2 x)$

Step 5.1.3.3

Multiply $2$ by $3$ .

$0 + 6 x$

Step 5.1.4

Add $0$ and $6 x$ .

$6 x$

Step 5.2

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

$6 \int \frac{1}{u^{4}} \cdot \frac{1}{6} d u$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Multiply $\frac{1}{u^{4}}$ by $\frac{1}{6}$ .

$6 \int \frac{1}{u^{4} \cdot 6} d u$

Step 6.2

Move $6$ to the left of $u^{4}$ .

$6 \int \frac{1}{6 u^{4}} d u$

Step 7

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

$6 (\frac{1}{6} \int \frac{1}{u^{4}} d u)$

Step 8

Simplify the expression.

Tap for more steps...

Step 8.1

Simplify.

Tap for more steps...

Step 8.1.1

Combine $\frac{1}{6}$ and $6$ .

$\frac{6}{6} \int \frac{1}{u^{4}} d u$

Step 8.1.2

Cancel the common factor of $6$ .

Tap for more steps...

Step 8.1.2.1

Cancel the common factor.

$\frac{6}{6} \int \frac{1}{u^{4}} d u$

Step 8.1.2.2

Rewrite the expression.

$1 \int \frac{1}{u^{4}} d u$

Step 8.1.3

Multiply $\int \frac{1}{u^{4}} d u$ by $1$ .

$\int \frac{1}{u^{4}} d u$

Step 8.2

Apply basic rules of exponents.

Tap for more steps...

Step 8.2.1

Move $u^{4}$ out of the denominator by raising it to the $- 1$ power.

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

Step 8.2.2

Multiply the exponents in ${(u^{4})}^{- 1}$ .

Tap for more steps...

Step 8.2.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int u^{4 \cdot - 1} d u$

Step 8.2.2.2

Multiply $4$ by $- 1$ .

$\int u^{- 4} d u$

Step 9

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

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

Step 10

Simplify.

Tap for more steps...

Step 10.1

Rewrite $- \frac{1}{3} u^{- 3} + C$ as $- \frac{1}{3} \cdot \frac{1}{u^{3}} + C$ .

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

Step 10.2

Simplify.

Tap for more steps...

Step 10.2.1

Multiply $\frac{1}{u^{3}}$ by $\frac{1}{3}$ .

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

Step 10.2.2

Move $3$ to the left of $u^{3}$ .

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

Step 11

Replace all occurrences of $u$ with $5 + 3 x^{2}$ .

$- \frac{1}{3 {(5 + 3 x^{2})}^{3}} + C$

Step 12

The answer is the antiderivative of the function $f (x) = \frac{6 x}{{(5 + 3 x^{2})}^{4}}$ .

$F (x) =$ $- \frac{1}{3 {(5 + 3 x^{2})}^{3}} + C$