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

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

Popular >

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

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

- ln ∣ x ∣ + ln x^{2} + 1 + C

Solution steps

\int \frac{x ^{2} - 1}{x ^{3} + x} d x

Take the partial fraction of

\frac{x ^{2} - 1}{x ^{3} + x} : - \frac{1}{x} + \frac{2 x}{x ^{2} + 1}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int \frac{1}{x} d x = ln ∣ x ∣

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

= - ln ∣ x ∣ + ln x^{2} + 1

Add a constant to the solution

= - ln ∣ x ∣ + ln x^{2} + 1 + C

\frac{d}{d x} (- k e^{l (x - m)} + n e^{- p (x - m)} + q)

\frac{1}{( s - 2 ) ^{2} + 9}

\frac{d}{d j} (6 i + 10 j)

(x + y) d x - (x + y - 2) d y = 0

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

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