Calculus Examples

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

Step 1

Split the fraction into multiple fractions.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

Simplify.

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Step 3.1

Cancel the common factor of $x^{2}$ and $x$ .

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Step 3.1.1

Factor $x$ out of $x^{2}$ .

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

Step 3.1.2

Cancel the common factors.

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Step 3.1.2.1

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

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

Step 3.1.2.2

Factor $x$ out of $x^{1}$ .

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

Step 3.1.2.3

Cancel the common factor.

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

Step 3.1.2.4

Rewrite the expression.

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

Step 3.1.2.5

Divide $x$ by $1$ .

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

Step 3.2

Move the negative in front of the fraction.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

Substitute and simplify.

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Step 7.1.1

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

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

Step 7.1.2

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

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

Step 7.1.3

Simplify.

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Step 7.1.3.1

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

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

Step 7.1.3.2

One to any power is one.

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

Step 7.1.3.3

Multiply $- 1$ by $1$ .

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

Step 7.1.3.4

To write $- (\ln (| e |) - \ln (| 1 |))$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 7.1.3.5

Combine $- (\ln (| e |) - \ln (| 1 |))$ and $\frac{2}{2}$ .

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

Step 7.1.3.6

Combine the numerators over the common denominator.

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

Step 7.1.3.7

Multiply $2$ by $- 1$ .

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

Step 7.2

Simplify.

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Step 7.2.1

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

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

Step 7.2.2

Combine the numerators over the common denominator.

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

Step 7.3

Simplify.

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Step 7.3.1

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

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

Step 7.3.2

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

$\frac{e^{2} - 2 \ln (\frac{e}{1}) - 1}{2}$

Step 7.3.3

Divide $e$ by $1$ .

$\frac{e^{2} - 2 \ln (e) - 1}{2}$

Step 7.3.4

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

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

Step 7.3.5

Multiply $- 2$ by $1$ .

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

Step 7.3.6

Subtract $1$ from $- 2$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$2.19452804 \dots$

Step 9