Calculus Examples

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

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

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

Step 3

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

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Step 3.1

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

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Step 3.1.1

Differentiate $x + 1$ .

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

Step 3.1.2

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

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

Step 3.1.3

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

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

Step 3.1.4

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

$1 + 0$

Step 3.1.5

Add $1$ and $0$ .

$1$

Step 3.2

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

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

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 5

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

$- u^{- 1} + C$

Step 6

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

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

Step 7

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

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

Step 8

The answer is the antiderivative of the function $f (x) = \frac{1}{{(x + 1)}^{2}}$ .

$F (x) =$ $- \frac{1}{x + 1} + C$