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

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

Popular >

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

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

- ln ∣ x ∣ + ln ∣ x - 1 ∣ + x + C

Solution steps

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

\frac{x ^{3} + 1}{x ^{3} - x} = \frac{1}{x ( x - 1 )} + 1

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

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

\int 1 d x = x

= - ln ∣ x ∣ + ln ∣ x - 1 ∣ + x

Add a constant to the solution

= - ln ∣ x ∣ + ln ∣ x - 1 ∣ + x + C

\int \frac{( x + 2 ) ^{7}}{2 x} d x

\int 2 x cos (n x) d x

\int \frac{5 x - 3}{x ^{2} + 6 x + 5} d x

\int 10 e^{x + 3} d x

\int (cot (x) - 1)^{2} d x

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