Calculus Examples

$\int_{1}^{e} \frac{\sqrt{\ln (x)}}{x} d x$

Step 1

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 1.1

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

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Step 1.1.1

Differentiate $\ln (x)$ .

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

Step 1.1.2

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

$\frac{1}{x}$

Step 1.2

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

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

Step 1.3

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

$u_{lower} = 0$

Step 1.4

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

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

Step 1.5

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

$u_{upper} = 1$

Step 1.6

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

$u_{lower} = 0$

$u_{upper} = 1$

Step 1.7

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

$\int_{0}^{1} \sqrt{u} d u$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$\int_{0}^{1} u^{\frac{1}{2}} d u$

Step 3

By the Power Rule, the integral of $u^{\frac{1}{2}}$ with respect to $u$ is $\frac{2}{3} u^{\frac{3}{2}}$ .

$\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1}$

Step 4

Substitute and simplify.

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Step 4.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $1$ and at $0$ .

$(\frac{2}{3} \cdot 1^{\frac{3}{2}}) - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2

Simplify.

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Step 4.2.1

One to any power is one.

$\frac{2}{3} \cdot 1 - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.2

Multiply $\frac{2}{3}$ by $1$ .

$\frac{2}{3} - \frac{2}{3} \cdot 0^{\frac{3}{2}}$

Step 4.2.3

Rewrite $0$ as $0^{2}$ .

$\frac{2}{3} - \frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}}$

Step 4.2.4

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}$

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Cancel the common factor.

$\frac{2}{3} - \frac{2}{3} \cdot 0^{2 (\frac{3}{2})}$

Step 4.2.5.2

Rewrite the expression.

$\frac{2}{3} - \frac{2}{3} \cdot 0^{3}$

Step 4.2.6

Raising $0$ to any positive power yields $0$ .

$\frac{2}{3} - \frac{2}{3} \cdot 0$

Step 4.2.7

Multiply $0$ by $- 1$ .

$\frac{2}{3} + 0 (\frac{2}{3})$

Step 4.2.8

Multiply $0$ by $\frac{2}{3}$ .

$\frac{2}{3} + 0$

Step 4.2.9

Add $\frac{2}{3}$ and $0$ .

$\frac{2}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$\frac{2}{3}$

Decimal Form:

$0.\overline{6}$