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

The integral of 1/((x-2)^2(x^2+3)^2) is

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

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

\frac{1}{147 3} x sin (2 arctan (\frac{x}{3})) - \frac{8}{343} ln ∣ x - 2 ∣ + \frac{4}{343} ln x^{2} + 3 + \frac{61}{2058 3} arctan (\frac{x}{3}) + \frac{1}{588 3} sin (2 arctan (\frac{x}{3})) - \frac{1}{49 ( x - 2 )} + C

Solution steps

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

Take the partial fraction of

\frac{1}{( x - 2 ) ^{2} ( x ^{2} + 3 ) ^{2}} : - \frac{8}{343 ( x - 2 )} + \frac{1}{49 ( x - 2 ) ^{2}} + \frac{8 x + 9}{343 ( x ^{2} + 3 )} + \frac{4 x + 1}{49 ( x ^{2} + 3 ) ^{2}}

= \int - \frac{8}{343 ( x - 2 )} + \frac{1}{49 ( x - 2 ) ^{2}} + \frac{8 x + 9}{343 ( x ^{2} + 3 )} + \frac{4 x + 1}{49 ( x ^{2} + 3 ) ^{2}} 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 \frac{8}{343 ( x - 2 )} d x + \int \frac{1}{49 ( x - 2 ) ^{2}} d x + \int \frac{8 x + 9}{343 ( x ^{2} + 3 )} d x + \int \frac{4 x + 1}{49 ( x ^{2} + 3 ) ^{2}} d x

\int \frac{8}{343 ( x - 2 )} d x = \frac{8}{343} ln ∣ x - 2 ∣

\int \frac{1}{49 ( x - 2 ) ^{2}} d x = - \frac{1}{49 ( x - 2 )}

\int \frac{8 x + 9}{343 ( x ^{2} + 3 )} d x = \frac{1}{343} (4 ln x^{2} + 3 + 33 arctan (\frac{x}{3}))

\int \frac{4 x + 1}{49 ( x ^{2} + 3 ) ^{2}} d x = \frac{1}{588 3} (4 x sin (2 arctan (\frac{x}{3})) + 2 arctan (\frac{x}{3}) + sin (2 arctan (\frac{x}{3})))

= - \frac{8}{343} ln ∣ x - 2 ∣ - \frac{1}{49 ( x - 2 )} + \frac{1}{343} (4 ln x^{2} + 3 + 33 arctan (\frac{x}{3})) + \frac{1}{588 3} (4 x sin (2 arctan (\frac{x}{3})) + 2 arctan (\frac{x}{3}) + sin (2 arctan (\frac{x}{3})))

Simplify

- \frac{8}{343} ln ∣ x - 2 ∣ - \frac{1}{49 ( x - 2 )} + \frac{1}{343} (4 ln x^{2} + 3 + 33 arctan (\frac{x}{3})) + \frac{1}{588 3} (4 x sin (2 arctan (\frac{x}{3})) + 2 arctan (\frac{x}{3}) + sin (2 arctan (\frac{x}{3}))) : \frac{1}{147 3} x sin (2 arctan (\frac{x}{3})) - \frac{8}{343} ln ∣ x - 2 ∣ + \frac{4}{343} ln x^{2} + 3 + \frac{61}{2058 3} arctan (\frac{x}{3}) + \frac{1}{588 3} sin (2 arctan (\frac{x}{3})) - \frac{1}{49 ( x - 2 )}

= \frac{1}{147 3} x sin (2 arctan (\frac{x}{3})) - \frac{8}{343} ln ∣ x - 2 ∣ + \frac{4}{343} ln x^{2} + 3 + \frac{61}{2058 3} arctan (\frac{x}{3}) + \frac{1}{588 3} sin (2 arctan (\frac{x}{3})) - \frac{1}{49 ( x - 2 )}

Add a constant to the solution

= \frac{1}{147 3} x sin (2 arctan (\frac{x}{3})) - \frac{8}{343} ln ∣ x - 2 ∣ + \frac{4}{343} ln x^{2} + 3 + \frac{61}{2058 3} arctan (\frac{x}{3}) + \frac{1}{588 3} sin (2 arctan (\frac{x}{3})) - \frac{1}{49 ( x - 2 )} + C

\frac{d}{d x} (e^{- \frac{x}{2}} cos (\frac{3}{2} x))

\frac{d}{d x} (- \frac{3}{2} x^{4} cos (x))

x^{2}, x

\int_{0}^{5} x^{3} d x

y^{^{''}} x^{2} - 5 x y^{^{'}} + 9 y = 0

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