Calculus Examples

$\int \frac{\ln^{2} (x)}{x^{3}} d x$

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.

$\int \ln^{2} (x) {(x^{3})}^{- 1} d x$

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, ${(a^{m})}^{n} = a^{m n}$ .

$\int \ln^{2} (x) x^{3 \cdot - 1} d x$

Step 1.2.2

Multiply $3$ by $- 1$ .

$\int \ln^{2} (x) x^{- 3} d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln^{2} (x)$ and $d v = x^{- 3}$ .

$\ln^{2} (x) (- \frac{1}{2 x^{2}}) - \int - \frac{1}{2 x^{2}} \cdot \frac{\ln (x^{2})}{x} d x$

Step 3

Simplify.

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Step 3.1

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \int - \frac{1}{2 x^{2}} \cdot \frac{\ln (x^{2})}{x} d x$

Step 3.2

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \int - \frac{\ln (x^{2})}{x (2 x^{2})} d x$

Step 3.3

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \int - \frac{\ln (x^{2})}{2 (x^{1} x^{2})} d x$

Step 3.4

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \int - \frac{\ln (x^{2})}{2 x^{1 + 2}} d x$

Step 3.5

Add $1$ and $2$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} - \int - \frac{\ln (x^{2})}{2 x^{3}} d x$

Step 4

Rewrite $- \frac{\ln (x^{2})}{2 x^{3}}$ as $- \frac{2 \ln (x)}{2 x^{3}}$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} - \int - \frac{2 \ln (x)}{2 x^{3}} d x$

Step 5

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - - \int \frac{2 \ln (x)}{2 x^{3}} d x$

Step 6

Simplify the expression.

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Step 6.1

Simplify.

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Step 6.1.1

Cancel the common factor of $2$ .

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Step 6.1.1.1

Cancel the common factor.

$- \frac{\ln^{2} (x)}{2 x^{2}} - - \int \frac{2 \ln (x)}{2 x^{3}} d x$

Step 6.1.1.2

Rewrite the expression.

$- \frac{\ln^{2} (x)}{2 x^{2}} - - \int \frac{\ln (x)}{x^{3}} d x$

Step 6.1.2

Multiply $- 1$ by $- 1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} + 1 \int \frac{\ln (x)}{x^{3}} d x$

Step 6.1.3

Multiply $\int \frac{\ln (x)}{x^{3}} d x$ by $1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} + \int \frac{\ln (x)}{x^{3}} d x$

Step 6.2

Apply basic rules of exponents.

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Step 6.2.1

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

$- \frac{\ln^{2} (x)}{2 x^{2}} + \int \ln (x) {(x^{3})}^{- 1} d x$

Step 6.2.2

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

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Step 6.2.2.1

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

$- \frac{\ln^{2} (x)}{2 x^{2}} + \int \ln (x) x^{3 \cdot - 1} d x$

Step 6.2.2.2

Multiply $3$ by $- 1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} + \int \ln (x) x^{- 3} d x$

Step 7

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

$- \frac{\ln^{2} (x)}{2 x^{2}} + \ln (x) (- \frac{1}{2 x^{2}}) - \int - \frac{1}{2 x^{2}} \cdot \frac{1}{x} d x$

Step 8

Simplify.

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Step 8.1

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} - \int - \frac{1}{2 x^{2}} \cdot \frac{1}{x} d x$

Step 8.2

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} - \int - \frac{1}{x (2 x^{2})} d x$

Step 8.3

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} - \int - \frac{1}{2 (x^{1} x^{2})} d x$

Step 8.4

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} - \int - \frac{1}{2 x^{1 + 2}} d x$

Step 8.5

Add $1$ and $2$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} - \int - \frac{1}{2 x^{3}} d x$

Step 9

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} - - \int \frac{1}{2 x^{3}} d x$

Step 10

Simplify.

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Step 10.1

Multiply $- 1$ by $- 1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} + 1 \int \frac{1}{2 x^{3}} d x$

Step 10.2

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} + \int \frac{1}{2 x^{3}} d x$

Step 11

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} + \frac{1}{2} \int \frac{1}{x^{3}} d x$

Step 12

Apply basic rules of exponents.

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Step 12.1

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} + \frac{1}{2} \int {(x^{3})}^{- 1} d x$

Step 12.2

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

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Step 12.2.1

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

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} + \frac{1}{2} \int x^{3 \cdot - 1} d x$

Step 12.2.2

Multiply $3$ by $- 1$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} + \frac{1}{2} \int x^{- 3} d x$

Step 13

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

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

Step 14

Simplify the answer.

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Step 14.1

Rewrite $- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} + \frac{1}{2} (- \frac{1}{2} x^{- 2} + C)$ as $- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} + \frac{1}{2} (- \frac{1}{2} \cdot \frac{1}{x^{2}}) + C$ .

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

Step 14.2

Simplify.

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Step 14.2.1

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

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

Step 14.2.2

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

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

Step 14.2.3

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

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

Step 14.2.4

Multiply $2$ by $2$ .

$- \frac{\ln^{2} (x)}{2 x^{2}} - \frac{\ln (x)}{2 x^{2}} - \frac{1}{4 x^{2}} + C$