Calculus Examples

$\int \frac{x}{{(7 x^{2} + 3)}^{5}} d x d x$

Step 1

Let $u = 7 x^{2} + 3$ . Then $d u = 14 x d x$ , so $\frac{1}{14} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 7 x^{2} + 3$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $7 x^{2} + 3$ .

$\frac{d}{d x} [7 x^{2} + 3]$

Step 1.1.2

By the Sum Rule, the derivative of $7 x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [3]$ .

$\frac{d}{d x} [7 x^{2}] + \frac{d}{d x} [3]$

Step 1.1.3

Evaluate $\frac{d}{d x} [7 x^{2}]$ .

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Step 1.1.3.1

Since $7$ is constant with respect to $x$ , the derivative of $7 x^{2}$ with respect to $x$ is $7 \frac{d}{d x} [x^{2}]$ .

$7 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$

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 $n = 2$ .

$7 (2 x) + \frac{d}{d x} [3]$

Step 1.1.3.3

Multiply $2$ by $7$ .

$14 x + \frac{d}{d x} [3]$

Step 1.1.4

Differentiate using the Constant Rule.

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Step 1.1.4.1

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$14 x + 0$

Step 1.1.4.2

Add $14 x$ and $0$ .

$14 x$

Step 1.2

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

$\int \frac{1}{u^{5}} \cdot \frac{1}{14} d u d x$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{1}{u^{5}}$ by $\frac{1}{14}$ .

$\int \frac{1}{u^{5} \cdot 14} d u d x$

Step 2.2

Move $14$ to the left of $u^{5}$ .

$\int \frac{1}{14 u^{5}} d u d x$

Step 3

Since $\frac{1}{14}$ is constant with respect to $u$ , move $\frac{1}{14}$ out of the integral.

$\frac{1}{14} \int \frac{1}{u^{5}} d u d x$

Step 4

Simplify the expression.

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Step 4.1

Simplify.

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Step 4.1.1

Combine $d$ and $\frac{1}{14}$ .

$\frac{d}{14} \int \frac{1}{u^{5}} d u x$

Step 4.1.2

Combine $x$ and $\frac{d}{14}$ .

$\frac{x d}{14} \int \frac{1}{u^{5}} d u$

Step 4.2

Apply basic rules of exponents.

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Step 4.2.1

Move $u^{5}$ out of the denominator by raising it to the $- 1$ power.

$\frac{x d}{14} \int {(u^{5})}^{- 1} d u$

Step 4.2.2

Multiply the exponents in ${(u^{5})}^{- 1}$ .

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Step 4.2.2.1

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

$\frac{x d}{14} \int u^{5 \cdot - 1} d u$

Step 4.2.2.2

Multiply $5$ by $- 1$ .

$\frac{x d}{14} \int u^{- 5} d u$

Step 5

By the Power Rule, the integral of $u^{- 5}$ with respect to $u$ is $- \frac{1}{4} u^{- 4}$ .

$\frac{x d}{14} (- \frac{1}{4} u^{- 4} + C)$

Step 6

Simplify.

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Step 6.1

Simplify.

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Step 6.1.1

Combine $u^{- 4}$ and $\frac{1}{4}$ .

$\frac{x d}{14} (- \frac{u^{- 4}}{4} + C)$

Step 6.1.2

Move $u^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{x d}{14} (- \frac{1}{4 u^{4}} + C)$

Step 6.2

Rewrite $\frac{x d}{14} (- \frac{1}{4 u^{4}} + C)$ as $\frac{1}{14} x d (- \frac{1}{4 u^{4}}) + C$ .

$\frac{1}{14} x d (- \frac{1}{4 u^{4}}) + C$

Step 6.3

Simplify.

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Step 6.3.1

Combine $\frac{1}{14}$ and $x$ .

$\frac{x}{14} d (- \frac{1}{4 u^{4}}) + C$

Step 6.3.2

Combine $\frac{x}{14}$ and $d$ .

$\frac{x d}{14} (- \frac{1}{4 u^{4}}) + C$

Step 6.3.3

Multiply $\frac{x d}{14}$ by $\frac{1}{4 u^{4}}$ .

$- \frac{x d}{14 (4 u^{4})} + C$

Step 6.3.4

Multiply $4$ by $14$ .

$- \frac{x d}{56 u^{4}} + C$

Step 7

Replace all occurrences of $u$ with $7 x^{2} + 3$ .

$- \frac{x d}{56 {(7 x^{2} + 3)}^{4}} + C$