What is the integral of (ln(1+x^2))/(x^3) ?

The integral of (ln(1+x^2))/(x^3) is -(ln(1+x^2))/(2x^2)+ln|x|-1/2 ln|x^2+1|+C

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integral of (ln(1+x^2))/(x^3)

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (ln(1+x^2))/(x^3) ?
The integral of (ln(1+x^2))/(x^3) is -(ln(1+x^2))/(2x^2)+ln|x|-1/2 ln|x^2+1|+C