Calculus Examples

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

Step 1

Let $u = 1 + x$ . Then $d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

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

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Step 1.1.1

Differentiate $1 + x$ .

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

Step 1.1.2

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

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

Step 1.1.3

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

$0 + \frac{d}{d x} [x]$

Step 1.1.4

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

$0 + 1$

Step 1.1.5

Add $0$ and $1$ .

$1$

Step 1.2

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Multiply $2$ by $- 1$ .

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

Step 3

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

$- u^{- 1} + C$

Step 4

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

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

Step 5

Replace all occurrences of $u$ with $1 + x$ .

$- \frac{1}{1 + x} + C$