What is the integral of (3x^3+2x+7)/(x^2+3) ?

The integral of (3x^3+2x+7)/(x^2+3) is 3/2 x^2+9/2-7/2 ln|x^2+3|+7/(sqrt(3))arctan(x/(sqrt(3)))+C

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integral of (3x^3+2x+7)/(x^2+3)

\int \frac{3 x ^{3} + 2 x + 7}{x ^{2} + 3} d x

\frac{3}{2} x^{2} + \frac{9}{2} - \frac{7}{2} ln x^{2} + 3 + \frac{7}{3} arctan (\frac{x}{3}) + C

Solution steps

\int \frac{3 x ^{3} + 2 x + 7}{x ^{2} + 3} d x

Expand

\frac{3 x ^{3} + 2 x + 7}{x ^{2} + 3} : \frac{3 x ^{3}}{x ^{2} + 3} + \frac{2 x}{x ^{2} + 3} + \frac{7}{x ^{2} + 3}

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{3 x ^{3}}{x ^{2} + 3} d x + \int \frac{2 x}{x ^{2} + 3} d x + \int \frac{7}{x ^{2} + 3} d x

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

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

\int \frac{7}{x ^{2} + 3} d x = \frac{7}{3} arctan (\frac{x}{3})

= \frac{3}{2} (x^{2} + 3 - 3 ln x^{2} + 3) + ln x^{2} + 3 + \frac{7}{3} arctan (\frac{x}{3})

Simplify

\frac{3}{2} (x^{2} + 3 - 3 ln x^{2} + 3) + ln x^{2} + 3 + \frac{7}{3} arctan (\frac{x}{3}) : \frac{3}{2} x^{2} + \frac{9}{2} - \frac{7}{2} ln x^{2} + 3 + \frac{7}{3} arctan (\frac{x}{3})

= \frac{3}{2} x^{2} + \frac{9}{2} - \frac{7}{2} ln x^{2} + 3 + \frac{7}{3} arctan (\frac{x}{3})

Add a constant to the solution

= \frac{3}{2} x^{2} + \frac{9}{2} - \frac{7}{2} ln x^{2} + 3 + \frac{7}{3} arctan (\frac{x}{3}) + C

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What is the integral of (3x^3+2x+7)/(x^2+3) ?
The integral of (3x^3+2x+7)/(x^2+3) is 3/2 x^2+9/2-7/2 ln|x^2+3|+7/(sqrt(3))arctan(x/(sqrt(3)))+C