Algebra Examples

$\ln^{2} (x)$

Step 1

Write $\ln^{2} (x)$ as a function.

$f (x) = \ln^{2} (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

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

Step 4

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

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

Step 5

Simplify.

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

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

Cancel the common factor of $x$ .

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Cancel the common factor.

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

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

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

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

Step 6

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

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

Step 7

Multiply $2$ by $- 1$ .

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

Step 8

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

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

Step 9

Simplify.

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

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

Cancel the common factor of $x$ .

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Cancel the common factor.

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

Rewrite the expression.

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

Step 10

Apply the constant rule.

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

Step 11

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

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

Step 12

The answer is the antiderivative of the function $f (x) = \ln^{2} (x)$ .

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