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

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

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

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

\frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x + C

Solution steps

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

Simplify

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

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

\int x^{2} d x = \frac{x ^{3}}{3}

\int x d x = \frac{x ^{2}}{2}

\int 1 d x = x

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x + C

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