What is the integral of (-3x^3+2x^2-28x+10)/(x^2+9) ?

The integral of (-3x^3+2x^2-28x+10)/(x^2+9) is -3/2 x^2+2x-1/2 ln|x^2+9|-8/3 arctan(x/3)-27/2+C

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integral of (-3x^3+2x^2-28x+10)/(x^2+9)

\int \frac{- 3 x ^{3} + 2 x ^{2} - 28 x + 10}{x ^{2} + 9} d x

- \frac{3}{2} x^{2} + 2 x - \frac{1}{2} ln x^{2} + 9 - \frac{8}{3} arctan (\frac{x}{3}) - \frac{27}{2} + C

Solution steps

\int \frac{- 3 x ^{3} + 2 x ^{2} - 28 x + 10}{x ^{2} + 9} d x

Expand

\frac{- 3 x ^{3} + 2 x ^{2} - 28 x + 10}{x ^{2} + 9} : - \frac{3 x ^{3}}{x ^{2} + 9} + \frac{2 x ^{2}}{x ^{2} + 9} - \frac{28 x}{x ^{2} + 9} + \frac{10}{x ^{2} + 9}

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} + 9} d x + \int \frac{2 x ^{2}}{x ^{2} + 9} d x - \int \frac{28 x}{x ^{2} + 9} d x + \int \frac{10}{x ^{2} + 9} d x

\int \frac{3 x ^{3}}{x ^{2} + 9} d x = \frac{3}{2} (x^{2} + 9 - 9 ln x^{2} + 9)

\int \frac{2 x ^{2}}{x ^{2} + 9} d x = 6 (- arctan (\frac{x}{3}) + \frac{x}{3})

\int \frac{28 x}{x ^{2} + 9} d x = 14 ln x^{2} + 9

\int \frac{10}{x ^{2} + 9} d x = \frac{10}{3} arctan (\frac{x}{3})

= - \frac{3}{2} (x^{2} + 9 - 9 ln x^{2} + 9) + 6 (- arctan (\frac{x}{3}) + \frac{x}{3}) - 14 ln x^{2} + 9 + \frac{10}{3} arctan (\frac{x}{3})

Simplify

- \frac{3}{2} (x^{2} + 9 - 9 ln x^{2} + 9) + 6 (- arctan (\frac{x}{3}) + \frac{x}{3}) - 14 ln x^{2} + 9 + \frac{10}{3} arctan (\frac{x}{3}) : - \frac{3}{2} x^{2} + 2 x - \frac{1}{2} ln x^{2} + 9 - \frac{8}{3} arctan (\frac{x}{3}) - \frac{27}{2}

= - \frac{3}{2} x^{2} + 2 x - \frac{1}{2} ln x^{2} + 9 - \frac{8}{3} arctan (\frac{x}{3}) - \frac{27}{2}

Add a constant to the solution

= - \frac{3}{2} x^{2} + 2 x - \frac{1}{2} ln x^{2} + 9 - \frac{8}{3} arctan (\frac{x}{3}) - \frac{27}{2} + C

\frac{d}{d x} (y^{2} - x)

\int x^{4} (x^{5} - 5)^{3} d x

f (x) = lo g_{e} (x)

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

\frac{\partial}{\partial x} (x^{\frac{2}{3}} y^{\frac{1}{3}})

What is the integral of (-3x^3+2x^2-28x+10)/(x^2+9) ?
The integral of (-3x^3+2x^2-28x+10)/(x^2+9) is -3/2 x^2+2x-1/2 ln|x^2+9|-8/3 arctan(x/3)-27/2+C