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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

\frac{d}{d x} (cot^{4} (x))

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

\frac{d}{d x} (e^{(x y)})

\int x + 5 d x

\frac{- x ^{2}}{2}

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