Calculus Examples

$\int \ln^{2} (4 x) d x$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln^{2} (4 x)$ and $d v = 1$ .

$\ln^{2} (4 x) x - \int x \frac{\ln (16 x^{2})}{x} d x$

Step 2

Simplify.

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Step 2.1

Combine $x$ and $\frac{\ln (16 x^{2})}{x}$ .

$\ln^{2} (4 x) x - \int \frac{x \ln (16 x^{2})}{x} d x$

Step 2.2

Cancel the common factor of $x$ .

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Step 2.2.1

Cancel the common factor.

$\ln^{2} (4 x) x - \int \frac{x \ln (16 x^{2})}{x} d x$

Step 2.2.2

Divide $\ln (16 x^{2})$ by $1$ .

$\ln^{2} (4 x) x - \int \ln (16 x^{2}) d x$

Step 3

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = \ln (16 x^{2})$ and $d v = 1$ .

$\ln^{2} (4 x) x - (\ln (16 x^{2}) x - \int x \frac{2}{x} d x)$

Step 4

Simplify.

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Step 4.1

Combine $x$ and $\frac{2}{x}$ .

$\ln^{2} (4 x) x - (\ln (16 x^{2}) x - \int \frac{x \cdot 2}{x} d x)$

Step 4.2

Move $2$ to the left of $x$ .

$\ln^{2} (4 x) x - (\ln (16 x^{2}) x - \int \frac{2 \cdot x}{x} d x)$

Step 4.3

Cancel the common factor of $x$ .

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Step 4.3.1

Cancel the common factor.

$\ln^{2} (4 x) x - (\ln (16 x^{2}) x - \int \frac{2 x}{x} d x)$

Step 4.3.2

Divide $2$ by $1$ .

$\ln^{2} (4 x) x - (\ln (16 x^{2}) x - \int 2 d x)$

Step 5

Apply the constant rule.

$\ln^{2} (4 x) x - (\ln (16 x^{2}) x - (2 x + C))$

Step 6

Rewrite $\ln^{2} (4 x) x - (\ln (16 x^{2}) x - (2 x + C))$ as $\ln^{2} (4 x) x - (\ln (16 x^{2}) x - 2 x) + C$ .

$\ln^{2} (4 x) x - (\ln (16 x^{2}) x - 2 x) + C$