Calculus Examples

$\int \frac{x}{{(x^{2} + 1)}^{2}} d x$

Step 1

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

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Step 1.1

Let $u = x^{2} + 1$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $x^{2} + 1$ .

$\frac{d}{d x} [x^{2} + 1]$

Step 1.1.2

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$

Step 1.1.3

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

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

Step 1.1.4

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$2 x + 0$

Step 1.1.5

Add $2 x$ and $0$ .

$2 x$

Step 1.2

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

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

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

Step 4

Apply basic rules of exponents.

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Step 4.1

Move $u^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 4.2

Multiply the exponents in ${(u^{2})}^{- 1}$ .

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Step 4.2.1

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

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

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

$- \frac{1}{2 u} + C$

Step 7

Replace all occurrences of $u$ with $x^{2} + 1$ .

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