Calculus Examples

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

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 $u = \ln (x)$ .

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

Step 1.3

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

$u_{lower} = 1$

Step 1.4

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

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

Step 1.5

Simplify.

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Step 1.5.1

Use logarithm rules to move $2$ out of the exponent.

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

Step 1.5.2

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

$u_{upper} = 2 \cdot 1$

Step 1.5.3

Multiply $2$ by $1$ .

$u_{upper} = 2$

Step 1.6

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

$u_{lower} = 1$

$u_{upper} = 2$

Step 1.7

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

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

Step 2

Apply basic rules of exponents.

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Step 2.1

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

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

Step 2.2

Move $u^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.3

Multiply the exponents in ${(u^{\frac{1}{2}})}^{- 1}$ .

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Step 2.3.1

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

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

Step 2.3.2

Combine $\frac{1}{2}$ and $- 1$ .

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

Multiply $2$ by $2^{\frac{1}{2}}$ by adding the exponents.

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Step 4.2.1.1

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

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Step 4.2.1.1.1

Raise $2$ to the power of $1$ .

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

Step 4.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2^{1 + \frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}}$

Step 4.2.1.2

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

$2^{\frac{2}{2} + \frac{1}{2}} - 2 \cdot 1^{\frac{1}{2}}$

Step 4.2.1.3

Combine the numerators over the common denominator.

$2^{\frac{2 + 1}{2}} - 2 \cdot 1^{\frac{1}{2}}$

Step 4.2.1.4

Add $2$ and $1$ .

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

Step 4.2.2

One to any power is one.

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

Step 4.2.3

Multiply $- 2$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.82842712 \dots$