Calculus Examples

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

Step 1

Apply basic rules of exponents.

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Step 1.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 1.2.1

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

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

Step 1.2.2

Multiply $2$ by $- 1$ .

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

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = x^{- 2}$ .

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

Simplify the expression.

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Step 5.1

Simplify.

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Step 5.1.1

Multiply $- 1$ by $- 1$ .

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

Step 5.1.2

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

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

Step 5.2

Apply basic rules of exponents.

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Step 5.2.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

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

Step 5.2.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 5.2.2.1

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

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

Step 5.2.2.2

Multiply $2$ by $- 1$ .

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

Step 6

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

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

Step 7

Simplify the answer.

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Step 7.1

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

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

Step 7.2

Substitute and simplify.

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Step 7.2.1

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

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

Step 7.2.2

Simplify.

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Step 7.2.2.1

Divide $\ln (1)$ by $1$ .

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

Step 7.2.2.2

One to any power is one.

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

Step 7.2.2.3

Multiply $- 1$ by $1$ .

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

Step 7.2.2.4

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

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

Step 7.2.2.5

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

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

Step 7.2.2.6

Combine the numerators over the common denominator.

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

Step 7.2.2.7

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

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

Step 7.2.2.8

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

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

Step 7.2.2.9

Combine the numerators over the common denominator.

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

Step 7.2.2.10

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

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

Step 7.2.2.11

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

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

Step 7.2.2.12

Subtract $1$ from $1$ .

$\frac{- e^{0} - \ln (e) - (- \ln (1) - 1) e}{e}$

Step 7.2.2.13

Anything raised to $0$ is $1$ .

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

Step 7.2.2.14

Multiply $- 1$ by $1$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 7.3.2

Factor $- 1$ out of $- \ln (e)$ .

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

Step 7.3.3

Factor $- 1$ out of $- 1 (1) - (\ln (e))$ .

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

Step 7.3.4

Factor $- 1$ out of $- (- \ln (1) - 1) e$ .

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

Step 7.3.5

Factor $- 1$ out of $- 1 (1 + \ln (e)) - ((- \ln (1) - 1) e)$ .

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

Step 7.3.6

Move the negative in front of the fraction.

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

Step 7.4

Simplify.

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Step 7.4.1

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

$- \frac{1 + \ln (e) + (- 0 - 1) e}{e}$

Step 7.4.2

Multiply $- 1$ by $0$ .

$- \frac{1 + \ln (e) + (0 - 1) e}{e}$

Step 7.4.3

Subtract $1$ from $0$ .

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

Step 7.4.4

Rewrite $- 1 e$ as $- e$ .

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

Step 7.4.5

Simplify the numerator.

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Step 7.4.5.1

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

$- \frac{1 + 1 - e}{e}$

Step 7.4.5.2

Add $1$ and $1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$0.26424111 \dots$