What is the integral of (ln(x+4))/(x^2) ?

The integral of (ln(x+4))/(x^2) is -(ln(x+4))/x+1/4 ln|x|-1/4 ln|x+4|+C

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integral of (ln(x+4))/(x^2)

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

- \frac{ln ( x + 4 )}{x} + \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 4 ∣ + C

Solution steps

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

Apply Integration By Parts

: - \frac{ln ( x + 4 )}{x} - \int - \frac{1}{x ( x + 4 )} d x

= - \frac{ln ( x + 4 )}{x} - \int - \frac{1}{x ( x + 4 )} d x

\int - \frac{1}{x ( x + 4 )} d x = - \frac{1}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x + 4 ∣

= - \frac{ln ( x + 4 )}{x} - (- \frac{1}{4} ln ∣ x ∣ + \frac{1}{4} ln ∣ x + 4 ∣)

Simplify

= - \frac{ln ( x + 4 )}{x} + \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 4 ∣

Add a constant to the solution

= - \frac{ln ( x + 4 )}{x} + \frac{1}{4} ln ∣ x ∣ - \frac{1}{4} ln ∣ x + 4 ∣ + C

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What is the integral of (ln(x+4))/(x^2) ?
The integral of (ln(x+4))/(x^2) is -(ln(x+4))/x+1/4 ln|x|-1/4 ln|x+4|+C