What is the integral of (x^3+x+5)/(x^2+2x+2) ?

The integral of (x^3+x+5)/(x^2+2x+2) is (x^2)/2-2x+3/2 ln|x^2+2x+2|+6arctan(x+1)+C

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integral of (x^3+x+5)/(x^2+2x+2)

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

\frac{x ^{2}}{2} - 2 x + \frac{3}{2} ln x^{2} + 2 x + 2 + 6 arctan (x + 1) + C

Solution steps

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

Expand

\frac{x ^{3} + x + 5}{x ^{2} + 2 x + 2} : \frac{x ^{3}}{x ^{2} + 2 x + 2} + \frac{x}{x ^{2} + 2 x + 2} + \frac{5}{x ^{2} + 2 x + 2}

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} + 2 x + 2} d x + \int \frac{x}{x ^{2} + 2 x + 2} d x + \int \frac{5}{x ^{2} + 2 x + 2} d x

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

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

\int \frac{5}{x ^{2} + 2 x + 2} d x = 5 arctan (x + 1)

= \frac{x ^{2}}{2} - 2 x + ln x^{2} + 2 x + 2 + 2 arctan (x + 1) + \frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1) + 5 arctan (x + 1)

Simplify

\frac{x ^{2}}{2} - 2 x + ln x^{2} + 2 x + 2 + 2 arctan (x + 1) + \frac{1}{2} ln x^{2} + 2 x + 2 - arctan (x + 1) + 5 arctan (x + 1) : \frac{x ^{2}}{2} - 2 x + \frac{3}{2} ln x^{2} + 2 x + 2 + 6 arctan (x + 1)

= \frac{x ^{2}}{2} - 2 x + \frac{3}{2} ln x^{2} + 2 x + 2 + 6 arctan (x + 1)

Add a constant to the solution

= \frac{x ^{2}}{2} - 2 x + \frac{3}{2} ln x^{2} + 2 x + 2 + 6 arctan (x + 1) + C

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What is the integral of (x^3+x+5)/(x^2+2x+2) ?
The integral of (x^3+x+5)/(x^2+2x+2) is (x^2)/2-2x+3/2 ln|x^2+2x+2|+6arctan(x+1)+C