Calculus Examples

$\int_{1}^{e} 4 x - \frac{11}{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int_{1}^{e} 4 x d x + \int_{1}^{e} - \frac{11}{x} d x$

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int_{1}^{e} x d x + \int_{1}^{e} - \frac{11}{x} d x$

Step 3

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

$4 (\frac{1}{2} x^{2}]_{1}^{e}) + \int_{1}^{e} - \frac{11}{x} d x$

Step 4

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

$4 (\frac{x^{2}}{2}]_{1}^{e}) + \int_{1}^{e} - \frac{11}{x} d x$

Step 5

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

$4 (\frac{x^{2}}{2}]_{1}^{e}) - \int_{1}^{e} \frac{11}{x} d x$

Step 6

Since $11$ is constant with respect to $x$ , move $11$ out of the integral.

$4 (\frac{x^{2}}{2}]_{1}^{e}) - (11 \int_{1}^{e} \frac{1}{x} d x)$

Step 7

Multiply $11$ by $- 1$ .

$4 (\frac{x^{2}}{2}]_{1}^{e}) - 11 \int_{1}^{e} \frac{1}{x} d x$

Step 8

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

$4 (\frac{x^{2}}{2}]_{1}^{e}) - 11 (\ln (| x |)]_{1}^{e})$

Step 9

Simplify the answer.

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Step 9.1

Substitute and simplify.

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Step 9.1.1

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

$4 ((\frac{e^{2}}{2}) - \frac{1^{2}}{2}) - 11 (\ln (| x |)]_{1}^{e})$

Step 9.1.2

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

$4 (\frac{e^{2}}{2} - \frac{1^{2}}{2}) - 11 (\ln (| e |) - \ln (| 1 |))$

Step 9.1.3

One to any power is one.

$4 (\frac{e^{2}}{2} - \frac{1}{2}) - 11 (\ln (| e |) - \ln (| 1 |))$

Step 9.2

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

$4 (\frac{e^{2}}{2} - \frac{1}{2}) - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3

Simplify.

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Step 9.3.1

Apply the distributive property.

$4 \frac{e^{2}}{2} + 4 (- \frac{1}{2}) - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.2

Cancel the common factor of $2$ .

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Step 9.3.2.1

Factor $2$ out of $4$ .

$2 (2) \frac{e^{2}}{2} + 4 (- \frac{1}{2}) - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.2.2

Cancel the common factor.

$2 \cdot 2 \frac{e^{2}}{2} + 4 (- \frac{1}{2}) - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.2.3

Rewrite the expression.

$2 e^{2} + 4 (- \frac{1}{2}) - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.3

Cancel the common factor of $2$ .

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Step 9.3.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$2 e^{2} + 4 (\frac{- 1}{2}) - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.3.2

Factor $2$ out of $4$ .

$2 e^{2} + 2 (2) \frac{- 1}{2} - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.3.3

Cancel the common factor.

$2 e^{2} + 2 \cdot 2 \frac{- 1}{2} - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.3.4

Rewrite the expression.

$2 e^{2} + 2 \cdot - 1 - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.4

Multiply $2$ by $- 1$ .

$2 e^{2} - 2 - 11 \ln (\frac{| e |}{| 1 |})$

Step 9.3.5

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

$2 e^{2} - 2 - 11 \ln (\frac{e}{| 1 |})$

Step 9.3.6

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

$2 e^{2} - 2 - 11 \ln (\frac{e}{1})$

Step 9.3.7

Divide $e$ by $1$ .

$2 e^{2} - 2 - 11 \ln (e)$

Step 9.3.8

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

$2 e^{2} - 2 - 11 \cdot 1$

Step 9.3.9

Multiply $- 11$ by $1$ .

$2 e^{2} - 2 - 11$

Step 9.3.10

Subtract $11$ from $- 2$ .

$2 e^{2} - 13$

Step 10

The result can be shown in multiple forms.

Exact Form:

$2 e^{2} - 13$

Decimal Form:

$1.77811219 \dots$

Step 11