Calculus Examples

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

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 $u = \ln (x)$ .

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

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.

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

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 $u^{\frac{1}{2}}$ .

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

Step 2.2

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

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

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

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

Step 2.3.2

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

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

Step 2.3.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Substitute and simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

One to any power is one.

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

Step 4.2.2

Multiply $2$ by $1$ .

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Cancel the common factor of $2$ .

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Step 4.2.5.1

Cancel the common factor.

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

Step 4.2.5.2

Rewrite the expression.

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

Step 4.2.6

Evaluate the exponent.

$2 - 2 \cdot 0$

Step 4.2.7

Multiply $- 2$ by $0$ .

$2 + 0$

Step 4.2.8

Add $2$ and $0$ .

$2$