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

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

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

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

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

Solution steps

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

Expand

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

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

Add a constant to the solution

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

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\int \frac{cos ( x )}{sin ^{2} ( x ) sin ( x )} d x

\int \frac{x - 6}{x ^{2} - 18 x + 82} d x

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