Calculus Examples

$\int_{1}^{2} \frac{1}{x \ln ({(x)}^{p})} d x$

Step 1

Rewrite $\frac{1}{x \ln (x^{p})}$ as $\frac{1}{x (p \ln (x))}$ .

$\int_{1}^{2} \frac{1}{x (p \ln (x))} d x$

Step 2

Since $\frac{1}{p}$ is constant with respect to $x$ , move $\frac{1}{p}$ out of the integral.

$\frac{1}{p} \int_{1}^{2} \frac{1}{x (\ln (x))} d x$

Step 3

Let $u = \ln (x)$ . Then $d u = \frac{1}{x} d x$ , so $x d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = \ln (x)$ . Find $\frac{d u}{d x}$ .

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Step 3.1.1

Differentiate $\ln (x)$ .

$\frac{d}{d x} [\ln (x)]$

Step 3.1.2

The derivative of $\ln (x)$ with respect to $x$ is $\frac{1}{x}$ .

$\frac{1}{x}$

Step 3.2

Substitute the lower limit in for $x$ in $u = \ln (x)$ .

$u_{lower} = \ln (1)$

Step 3.3

The natural logarithm of $1$ is $0$ .

$u_{lower} = 0$

Step 3.4

Substitute the upper limit in for $x$ in $u = \ln (x)$ .

$u_{upper} = \ln (2)$

Step 3.5

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

$u_{lower} = 0$

$u_{upper} = \ln (2)$

Step 3.6

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

$\frac{1}{p} \int_{0}^{\ln (2)} \frac{1}{u} d u$

Step 4

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

$\frac{1}{p} \ln (| u |)]_{0}^{\ln (2)}$

Step 5

Combine $\frac{1}{p}$ and $\ln (| u |)]_{0}^{\ln (2)}$ .

$\frac{\ln (| u |)]_{0}^{\ln (2)}}{p}$

Step 6

Evaluate $\ln (| u |)$ at $\ln (2)$ and at $0$ .

$\frac{\ln (| \ln (2) |) - \ln (| 0 |)}{p}$

Step 7

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

$\frac{\ln (\frac{| \ln (2) |}{| 0 |})}{p}$

Step 8

Simplify.

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Step 8.1

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

$\frac{\ln (\frac{| \ln (2) |}{0})}{p}$

Step 8.2

The expression contains a division by $0$ . The expression is undefined.

Undefined

$\frac{\ln (\frac{| \ln (2) |}{0})}{p}$

Step 9

The expression contains a division by $0$ . The expression is undefined.

Undefined