What is the integral of (x^4+x^3)/(x^2+1) ?

The integral of (x^4+x^3)/(x^2+1) is (x^3)/3+(x^2)/2-x-1/2 ln|x^2+1|+arctan(x)+C

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integral of (x^4+x^3)/(x^2+1)

\int \frac{x ^{4} + x ^{3}}{x ^{2} + 1} d x

\frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x - \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

Solution steps

\int \frac{x ^{4} + x ^{3}}{x ^{2} + 1} d x

Long division

\frac{x ^{4} + x ^{3}}{x ^{2} + 1} : x^{2} + x - 1 + \frac{- x + 1}{x ^{2} + 1}

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

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

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

Add a constant to the solution

= \frac{x ^{3}}{3} + \frac{x ^{2}}{2} - x - \frac{1}{2} ln x^{2} + 1 + arctan (x) + C

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What is the integral of (x^4+x^3)/(x^2+1) ?
The integral of (x^4+x^3)/(x^2+1) is (x^3)/3+(x^2)/2-x-1/2 ln|x^2+1|+arctan(x)+C