Calculus Examples

$\int_{0}^{e} x + \frac{1}{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{0}^{e} x d x + \int_{0}^{e} \frac{1}{x} d x$

Step 2

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

$\frac{1}{2} x^{2}]_{0}^{e} + \int_{0}^{e} \frac{1}{x} d x$

Step 3

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

$\frac{1}{2} x^{2}]_{0}^{e} + \ln (| x |)]_{0}^{e}$

Step 4

Simplify the answer.

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Step 4.1

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

$\frac{1}{2} x^{2} + \ln (| x |)]_{0}^{e}$

Step 4.2

Substitute and simplify.

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Step 4.2.1

Evaluate $\frac{1}{2} x^{2} + \ln (| x |)$ at $e$ and at $0$ .

$(\frac{1}{2} e^{2} + \ln (| e |)) - (\frac{1}{2} \cdot 0^{2} + \ln (| 0 |))$

Step 4.2.2

Simplify.

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Step 4.2.2.1

Combine $\frac{1}{2}$ and $e^{2}$ .

$\frac{e^{2}}{2} + \ln (| e |) - (\frac{1}{2} \cdot 0^{2} + \ln (| 0 |))$

Step 4.2.2.2

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

$\frac{e^{2}}{2} + \ln (| e |) - (\frac{1}{2} \cdot 0 + \ln (| 0 |))$

Step 4.2.2.3

Multiply $\frac{1}{2}$ by $0$ .

$\frac{e^{2}}{2} + \ln (| e |) - (0 + \ln (| 0 |))$

Step 4.2.2.4

Add $0$ and $\ln (| 0 |)$ .

$\frac{e^{2}}{2} + \ln (| e |) - \ln (| 0 |)$

Step 4.3

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

$\frac{e^{2}}{2} + \ln (\frac{| e |}{| 0 |})$

Step 4.4

Simplify.

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Step 4.4.1

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

$\frac{e^{2}}{2} + \ln (\frac{| e |}{0})$

Step 4.4.2

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

Undefined

$\frac{e^{2}}{2} + \ln (\frac{| e |}{0})$

Step 4.5

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

Undefined

$\frac{e^{2}}{2} + \ln (\frac{| e |}{0})$

Step 5

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

Undefined