Calculus Examples

$\int 5 \ln (x^{2}) d x$

Step 1

Rewrite $5 \ln (x^{2})$ as $5 (2 \ln (x))$ .

$\int 5 (2 \ln (x)) d x$

Step 2

Since $5 \cdot 2$ is constant with respect to $x$ , move $5 \cdot 2$ out of the integral.

$5 \cdot 2 \int \ln (x) d x$

Step 3

Multiply $5$ by $2$ .

$10 \int \ln (x) d x$

Step 4

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (x)$ and $d v = 1$ .

$10 (\ln (x) x - \int x \frac{1}{x} d x)$

Step 5

Simplify.

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Step 5.1

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

$10 (\ln (x) x - \int \frac{x}{x} d x)$

Step 5.2

Cancel the common factor of $x$ .

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Step 5.2.1

Cancel the common factor.

$10 (\ln (x) x - \int \frac{x}{x} d x)$

Step 5.2.2

Rewrite the expression.

$10 (\ln (x) x - \int d x)$

Step 6

Apply the constant rule.

$10 (\ln (x) x - (x + C))$

Step 7

Simplify.

$10 (\ln (x) x - x) + C$