Calculus Examples

$\int \frac{2 x + 7}{{(2 x^{2} + 14 x)}^{4}} d x$

Step 1

Let $u = 2 x^{2} + 14 x$ . Then $d u = (4 x + 14) d x$ , so $\frac{1}{2} d u = (2 x + 7) d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 2 x^{2} + 14 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $2 x^{2} + 14 x$ .

$\frac{d}{d x} [2 x^{2} + 14 x]$

Step 1.1.2

By the Sum Rule, the derivative of $2 x^{2} + 14 x$ with respect to $x$ is $\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [14 x]$ .

$\frac{d}{d x} [2 x^{2}] + \frac{d}{d x} [14 x]$

Step 1.1.3

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

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Step 1.1.3.1

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

$2 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [14 x]$

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 $n = 2$ .

$2 (2 x) + \frac{d}{d x} [14 x]$

Step 1.1.3.3

Multiply $2$ by $2$ .

$4 x + \frac{d}{d x} [14 x]$

Step 1.1.4

Evaluate $\frac{d}{d x} [14 x]$ .

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Step 1.1.4.1

Since $14$ is constant with respect to $x$ , the derivative of $14 x$ with respect to $x$ is $14 \frac{d}{d x} [x]$ .

$4 x + 14 \frac{d}{d x} [x]$

Step 1.1.4.2

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

$4 x + 14 \cdot 1$

Step 1.1.4.3

Multiply $14$ by $1$ .

$4 x + 14$

Step 1.2

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

$\int \frac{1}{u^{4}} \cdot \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

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

$\int \frac{1}{u^{4} \cdot 2} d u$

Step 2.2

Move $2$ to the left of $u^{4}$ .

$\int \frac{1}{2 u^{4}} d u$

Step 3

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

$\frac{1}{2} \int \frac{1}{u^{4}} d u$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$\frac{1}{2} \int {(u^{4})}^{- 1} d u$

Step 4.2

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

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Step 4.2.1

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

$\frac{1}{2} \int u^{4 \cdot - 1} d u$

Step 4.2.2

Multiply $4$ by $- 1$ .

$\frac{1}{2} \int u^{- 4} d u$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{2} (- \frac{1}{3} u^{- 3} + C)$ as $\frac{1}{2} (- \frac{1}{3} \cdot \frac{1}{u^{3}}) + C$ .

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

Step 6.2

Simplify.

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Step 6.2.1

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

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

Step 6.2.2

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

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

Step 6.2.3

Multiply $\frac{1}{2}$ by $\frac{1}{3 u^{3}}$ .

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

Step 6.2.4

Multiply $3$ by $2$ .

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

Step 7

Replace all occurrences of $u$ with $2 x^{2} + 14 x$ .

$- \frac{1}{6 {(2 x^{2} + 14 x)}^{3}} + C$