What is the integral of (ln(x))/((1+2x)^2) ?

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

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integral of (ln(x))/((1+2x)^2)

integral

\frac{ln ( x )}{( 1 + 2 x ) ^{2}}

- \frac{ln ( x )}{2 ( 1 + 2 x )} + \frac{1}{2} (ln ∣ x ∣ - ln ∣ 2 x + 1 ∣) + C

Solution steps

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

Apply Integration By Parts

= - \frac{ln ( x )}{2 ( 1 + 2 x )} - \int - \frac{1}{2 x ( 1 + 2 x )} d x

\int - \frac{1}{2 x ( 1 + 2 x )} d x = - \frac{1}{2} (ln ∣ x ∣ - ln ∣ 2 x + 1 ∣)

= - \frac{ln ( x )}{2 ( 1 + 2 x )} - (- \frac{1}{2} (ln ∣ x ∣ - ln ∣ 2 x + 1 ∣))

Simplify

= - \frac{ln ( x )}{2 ( 1 + 2 x )} + \frac{1}{2} (ln ∣ x ∣ - ln ∣ 2 x + 1 ∣)

Add a constant to the solution

= - \frac{ln ( x )}{2 ( 1 + 2 x )} + \frac{1}{2} (ln ∣ x ∣ - ln ∣ 2 x + 1 ∣) + C

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