Calculus Examples

$\ln^{3} (x)$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln^{3} (x)$ and $d v = 1$ .

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

Step 2

Simplify.

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Combine $x$ and $\frac{3 \ln^{2} (x)}{x}$ .

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

Cancel the common factor of $x$ .

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Cancel the common factor.

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

Divide $3 \ln^{2} (x)$ by $1$ .

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

Step 3

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 4

Multiply $3$ by $- 1$ .

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

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln^{2} (x)$ and $d v = 1$ .

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

Step 6

Simplify.

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

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

Cancel the common factor of $x$ .

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Cancel the common factor.

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

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

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

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

Step 7

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

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

Step 8

Multiply $2$ by $- 1$ .

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

Step 9

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

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

Step 10

Simplify.

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

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

Cancel the common factor of $x$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Step 11

Apply the constant rule.

$\ln^{3} (x) x - 3 (\ln^{2} (x) x - 2 (\ln (x) x - (x + C)))$

Step 12

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

$\ln^{3} (x) x - 3 (\ln^{2} (x) x - 2 \ln (x) x + 2 x) + C$