Calculus Examples

$\int_{1}^{e} x^{2} - \frac{1}{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{e} x^{2} d x + \int_{1}^{e} - \frac{1}{x} d x$

Step 2

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

$\frac{1}{3} x^{3}]_{1}^{e} + \int_{1}^{e} - \frac{1}{x} d x$

Step 3

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{3} x^{3}]_{1}^{e} - \int_{1}^{e} \frac{1}{x} d x$

Step 4

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

$\frac{1}{3} x^{3}]_{1}^{e} - (\ln (| x |)]_{1}^{e})$

Step 5

Simplify the answer.

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Step 5.1

Substitute and simplify.

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Step 5.1.1

Evaluate $\frac{1}{3} x^{3}$ at $e$ and at $1$ .

$(\frac{1}{3} e^{3}) - \frac{1}{3} \cdot 1^{3} - (\ln (| x |)]_{1}^{e})$

Step 5.1.2

Evaluate $\ln (| x |)$ at $e$ and at $1$ .

$\frac{1}{3} e^{3} - \frac{1}{3} \cdot 1^{3} - (\ln (| e |) - \ln (| 1 |))$

Step 5.1.3

Simplify.

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Step 5.1.3.1

Combine $\frac{1}{3}$ and $e^{3}$ .

$\frac{e^{3}}{3} - \frac{1}{3} \cdot 1^{3} - (\ln (| e |) - \ln (| 1 |))$

Step 5.1.3.2

One to any power is one.

$\frac{e^{3}}{3} - \frac{1}{3} \cdot 1 - (\ln (| e |) - \ln (| 1 |))$

Step 5.1.3.3

Multiply $- 1$ by $1$ .

$\frac{e^{3}}{3} - \frac{1}{3} - (\ln (| e |) - \ln (| 1 |))$

Step 5.1.3.4

To write $- (\ln (| e |) - \ln (| 1 |))$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{e^{3}}{3} - (\ln (| e |) - \ln (| 1 |)) \cdot \frac{3}{3} - \frac{1}{3}$

Step 5.1.3.5

Combine $- (\ln (| e |) - \ln (| 1 |))$ and $\frac{3}{3}$ .

$\frac{e^{3}}{3} + \frac{- (\ln (| e |) - \ln (| 1 |)) \cdot 3}{3} - \frac{1}{3}$

Step 5.1.3.6

Combine the numerators over the common denominator.

$\frac{e^{3} - (\ln (| e |) - \ln (| 1 |)) \cdot 3}{3} - \frac{1}{3}$

Step 5.1.3.7

Multiply $3$ by $- 1$ .

$\frac{e^{3} - 3 (\ln (| e |) - \ln (| 1 |))}{3} - \frac{1}{3}$

Step 5.2

Simplify.

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Step 5.2.1

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

$\frac{e^{3} - 3 \ln (\frac{| e |}{| 1 |})}{3} - \frac{1}{3}$

Step 5.2.2

Combine the numerators over the common denominator.

$\frac{e^{3} - 3 \ln (\frac{| e |}{| 1 |}) - 1}{3}$

Step 5.3

Simplify.

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Step 5.3.1

$e$ is approximately $2.71828182$ which is positive so remove the absolute value

$\frac{e^{3} - 3 \ln (\frac{e}{| 1 |}) - 1}{3}$

Step 5.3.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$\frac{e^{3} - 3 \ln (\frac{e}{1}) - 1}{3}$

Step 5.3.3

Divide $e$ by $1$ .

$\frac{e^{3} - 3 \ln (e) - 1}{3}$

Step 5.3.4

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

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

Step 5.3.5

Multiply $- 3$ by $1$ .

$\frac{e^{3} - 3 - 1}{3}$

Step 5.3.6

Subtract $1$ from $- 3$ .

$\frac{e^{3} - 4}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{e^{3} - 4}{3}$

Decimal Form:

$5.36184564 \dots$

Step 7