Calculus Examples

$\int_{2}^{e + 1} \frac{4}{x - 1} d x$

Step 1

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{2}^{e + 1} \frac{1}{x - 1} d x$

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

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

$u_{lower} = 2 - 1$

Step 2.3

Subtract $1$ from $2$ .

$u_{lower} = 1$

Step 2.4

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

$u_{upper} = e + 1 - 1$

Step 2.5

Simplify.

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Step 2.5.1

Subtract $1$ from $1$ .

$u_{upper} = e + 0$

Step 2.5.2

Add $e$ and $0$ .

$u_{upper} = e$

Step 2.6

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

$u_{lower} = 1$

$u_{upper} = e$

Step 2.7

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

$4 \int_{1}^{e} \frac{1}{u} d u$

Step 3

The integral of $\frac{1}{u}$ with respect to $u$ is $\ln (| u |)$ .

$4 (\ln (| u |)]_{1}^{e})$

Step 4

Evaluate $\ln (| u |)$ at $e$ and at $1$ .

$4 (\ln (| e |) - \ln (| 1 |))$

Step 5

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$4 \ln (\frac{| e |}{| 1 |})$

Step 6

Simplify.

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Step 6.1

$e$ is approximately $2.71828182$ which is positive so remove the absolute value

$4 \ln (\frac{e}{| 1 |})$

Step 6.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$4 \ln (\frac{e}{1})$

Step 6.3

Divide $e$ by $1$ .

$4 \ln (e)$

Step 6.4

The natural logarithm of $e$ is $1$ .

$4 \cdot 1$

Step 6.5

Multiply $4$ by $1$ .

$4$

Step 7