Calculus Examples

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

Step 1

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

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

Step 2

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 3

Apply basic rules of exponents.

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Multiply $2$ by $- 1$ .

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

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

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

Step 5.5

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

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

Step 5.6

Add $1$ and $1$ .

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

Simplify.

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Step 7.1.1

Multiply $- 1$ by $- 1$ .

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

Step 7.1.2

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

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

Step 7.2

Apply basic rules of exponents.

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Step 7.2.1

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

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

Step 7.2.2

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

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Step 7.2.2.1

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

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

Step 7.2.2.2

Multiply $2$ by $- 1$ .

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

Step 8

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

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

Step 9

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

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