Calculus Examples

$\int_{1}^{e} \frac{\ln (x)}{x^{3}} d x$

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$\int_{1}^{e} \ln (x) {(x^{3})}^{- 1} d x$

Step 1.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 1.2.1

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

$\int_{1}^{e} \ln (x) x^{3 \cdot - 1} d x$

Step 1.2.2

Multiply $3$ by $- 1$ .

$\int_{1}^{e} \ln (x) x^{- 3} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{- 3}$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Multiply $\frac{1}{x}$ by $\frac{1}{2 x^{2}}$ .

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

Step 3.3

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

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

Step 3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.5

Add $1$ and $2$ .

$- \frac{\ln (x)}{2 x^{2}}]_{1}^{e} - \int_{1}^{e} - \frac{1}{2 x^{3}} d x$

Step 4

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

$- \frac{\ln (x)}{2 x^{2}}]_{1}^{e} - - \int_{1}^{e} \frac{1}{2 x^{3}} d x$

Step 5

Simplify.

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Step 5.1

Multiply $- 1$ by $- 1$ .

$- \frac{\ln (x)}{2 x^{2}}]_{1}^{e} + 1 \int_{1}^{e} \frac{1}{2 x^{3}} d x$

Step 5.2

Multiply $\int_{1}^{e} \frac{1}{2 x^{3}} d x$ by $1$ .

$- \frac{\ln (x)}{2 x^{2}}]_{1}^{e} + \int_{1}^{e} \frac{1}{2 x^{3}} d x$

Step 6

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$- \frac{\ln (x)}{2 x^{2}}]_{1}^{e} + \frac{1}{2} \int_{1}^{e} \frac{1}{x^{3}} d x$

Step 7

Apply basic rules of exponents.

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Step 7.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$- \frac{\ln (x)}{2 x^{2}}]_{1}^{e} + \frac{1}{2} \int_{1}^{e} {(x^{3})}^{- 1} d x$

Step 7.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 7.2.1

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

$- \frac{\ln (x)}{2 x^{2}}]_{1}^{e} + \frac{1}{2} \int_{1}^{e} x^{3 \cdot - 1} d x$

Step 7.2.2

Multiply $3$ by $- 1$ .

$- \frac{\ln (x)}{2 x^{2}}]_{1}^{e} + \frac{1}{2} \int_{1}^{e} x^{- 3} d x$

Step 8

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

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

Step 9

Substitute and simplify.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

Simplify.

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Step 9.3.1

One to any power is one.

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

Step 9.3.2

Multiply $2$ by $1$ .

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

Step 9.3.3

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

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

Step 9.3.4

Move $e^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9.3.5

One to any power is one.

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

Step 9.3.6

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

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

Step 9.3.7

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

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

Step 9.3.8

Combine $\frac{1}{2} (- \frac{1}{2 e^{2}} + \frac{1}{2})$ and $\frac{2 e^{2}}{2 e^{2}}$ .

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

Step 9.3.9

Combine the numerators over the common denominator.

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

Step 9.3.10

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

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

Step 9.3.11

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

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

Step 9.3.12

Move $2$ to the left of $e^{2}$ .

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

Step 9.3.13

Cancel the common factor of $2$ .

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Step 9.3.13.1

Cancel the common factor.

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

Step 9.3.13.2

Divide $e^{2}$ by $1$ .

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

Step 10

Simplify.

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Step 10.1

Simplify each term.

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Step 10.1.1

Simplify the numerator.

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Step 10.1.1.1

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

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

Step 10.1.1.2

Multiply $- 1$ by $1$ .

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

Step 10.1.1.3

Apply the distributive property.

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

Step 10.1.1.4

Cancel the common factor of $e^{2}$ .

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Step 10.1.1.4.1

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

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

Step 10.1.1.4.2

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

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

Step 10.1.1.4.3

Cancel the common factor.

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

Step 10.1.1.4.4

Rewrite the expression.

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

Step 10.1.1.5

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

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

Step 10.1.1.6

Move the negative in front of the fraction.

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

Step 10.1.1.7

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

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

Step 10.1.1.8

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

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

Step 10.1.1.9

Combine the numerators over the common denominator.

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

Step 10.1.1.10

Simplify the numerator.

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Step 10.1.1.10.1

Multiply $- 1$ by $2$ .

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

Step 10.1.1.10.2

Subtract $1$ from $- 2$ .

$\frac{\frac{- 3}{2} + \frac{e^{2}}{2}}{2 e^{2}} + \frac{\ln (1)}{2}$

Step 10.1.1.11

Combine the numerators over the common denominator.

$\frac{\frac{- 3 + e^{2}}{2}}{2 e^{2}} + \frac{\ln (1)}{2}$

Step 10.1.2

Multiply the numerator by the reciprocal of the denominator.

$\frac{- 3 + e^{2}}{2} \cdot \frac{1}{2 e^{2}} + \frac{\ln (1)}{2}$

Step 10.1.3

Multiply $\frac{- 3 + e^{2}}{2} \cdot \frac{1}{2 e^{2}}$ .

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Step 10.1.3.1

Multiply $\frac{- 3 + e^{2}}{2}$ by $\frac{1}{2 e^{2}}$ .

$\frac{- 3 + e^{2}}{2 (2 e^{2})} + \frac{\ln (1)}{2}$

Step 10.1.3.2

Multiply $2$ by $2$ .

$\frac{- 3 + e^{2}}{4 e^{2}} + \frac{\ln (1)}{2}$

Step 10.1.4

The natural logarithm of $1$ is $0$ .

$\frac{- 3 + e^{2}}{4 e^{2}} + \frac{0}{2}$

Step 10.1.5

Divide $0$ by $2$ .

$\frac{- 3 + e^{2}}{4 e^{2}} + 0$

Step 10.2

Add $\frac{- 3 + e^{2}}{4 e^{2}}$ and $0$ .

$\frac{- 3 + e^{2}}{4 e^{2}}$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\frac{- 3 + e^{2}}{4 e^{2}}$

Decimal Form:

$0.14849853 \dots$