Calculus Examples

$\int \frac{16}{x^{2} + 2 x + 1} d x$

Step 1

Factor using the perfect square rule.

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Step 1.1

Rewrite $1$ as $1^{2}$ .

$\int \frac{16}{x^{2} + 2 x + 1^{2}} d x$

Step 1.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 1.3

Rewrite the polynomial.

$\int \frac{16}{x^{2} + 2 \cdot x \cdot 1 + 1^{2}} d x$

Step 1.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

$\int \frac{16}{{(x + 1)}^{2}} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $x + 1$ .

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

Step 2.1.2

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 2.1.3

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 2.1.4

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

$1 + 0$

Step 2.1.5

Add $1$ and $0$ .

$1$

Step 2.2

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

$\int \frac{16}{u^{2}} d u$

Step 3

Since $16$ is constant with respect to $u$ , move $16$ out of the integral.

$16 \int \frac{1}{u^{2}} d u$

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.

$16 \int {(u^{2})}^{- 1} d u$

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

$16 \int u^{2 \cdot - 1} d u$

Step 4.2.2

Multiply $2$ by $- 1$ .

$16 \int u^{- 2} d u$

Step 5

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

$16 (- u^{- 1} + C)$

Step 6

Simplify.

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Step 6.1

Rewrite $16 (- u^{- 1} + C)$ as $16 (- \frac{1}{u}) + C$ .

$16 (- \frac{1}{u}) + C$

Step 6.2

Simplify.

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Step 6.2.1

Multiply $- 1$ by $16$ .

$- 16 \frac{1}{u} + C$

Step 6.2.2

Combine $- 16$ and $\frac{1}{u}$ .

$\frac{- 16}{u} + C$

Step 6.2.3

Move the negative in front of the fraction.

$- \frac{16}{u} + C$

Step 7

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

$- \frac{16}{x + 1} + C$