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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (x^{0.25} Y (x)^{0.5})

2 x^{2} - x - 3

\frac{1}{( s ^{2} + 0.2 s + 1 ) s}

x \to - 3 lim (x^{2} + 3 x + 1)

cos (π x) \frac{1}{2}

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