Calculus Examples

$\int_{0}^{a} \frac{1}{{(1 + x)}^{2}} d x = \frac{1}{6}$

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

Substitute the lower limit in for $x$ in $u = 1 + x$ .

$u_{lower} = 1 + 0$

Step 1.3

Add $1$ and $0$ .

$u_{lower} = 1$

Step 1.4

Substitute the upper limit in for $x$ in $u = 1 + x$ .

$u_{upper} = 1 + a$

Step 1.5

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 1 + a$

Step 1.6

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{1}^{1 + a} \frac{1}{u^{2}} d u$

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.

$\int_{1}^{1 + a} {(u^{2})}^{- 1} d u$

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

$\int_{1}^{1 + a} u^{2 \cdot - 1} d u$

Step 2.2.2

Multiply $2$ by $- 1$ .

$\int_{1}^{1 + a} u^{- 2} d u$

Step 3

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

$- u^{- 1}]_{1}^{1 + a}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $- u^{- 1}$ at $1 + a$ and at $1$ .

$(- {(1 + a)}^{- 1}) + 1^{- 1}$

Step 4.2

One to any power is one.

$- {(1 + a)}^{- 1} + 1$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- \frac{1}{1 + a} + 1$

Step 5.2

Write $1$ as a fraction with a common denominator.

$- \frac{1}{1 + a} + \frac{1 + a}{1 + a}$

Step 5.3

Combine the numerators over the common denominator.

$\frac{- 1 + 1 + a}{1 + a}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Add $- 1$ and $1$ .

$\frac{0 + a}{1 + a}$

Step 5.4.2

Add $0$ and $a$ .

$\frac{a}{1 + a}$