Calculus Examples

$\int \frac{2 x + 2}{{(x^{2} + 2 x - 4)}^{4}} d x$

Step 1

Let $u = x^{2} + 2 x - 4$ . Then $d u = (2 x + 2) d x$ , so $\frac{1}{2 x + 2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = x^{2} + 2 x - 4$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $x^{2} + 2 x - 4$ .

$\frac{d}{d x} [x^{2} + 2 x - 4]$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $x^{2} + 2 x - 4$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 4]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 4]$

Step 1.1.2.2

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

$2 x + \frac{d}{d x} [2 x] + \frac{d}{d x} [- 4]$

Step 1.1.3

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

Tap for more steps...

Step 1.1.3.1

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

$2 x + 2 \frac{d}{d x} [x] + \frac{d}{d x} [- 4]$

Step 1.1.3.2

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

$2 x + 2 \cdot 1 + \frac{d}{d x} [- 4]$

Step 1.1.3.3

Multiply $2$ by $1$ .

$2 x + 2 + \frac{d}{d x} [- 4]$

Step 1.1.4

Differentiate using the Constant Rule.

Tap for more steps...

Step 1.1.4.1

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4$ with respect to $x$ is $0$ .

$2 x + 2 + 0$

Step 1.1.4.2

Add $2 x + 2$ and $0$ .

$2 x + 2$

Step 1.2

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

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

Step 2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

Multiply $4$ by $- 1$ .

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

Step 3

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 4

Simplify.

Tap for more steps...

Step 4.1

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 4.2

Simplify.

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

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

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

Step 5

Replace all occurrences of $u$ with $x^{2} + 2 x - 4$ .

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