What is the integral of (8x^3+3)/((x^2+2)^2) ?

The integral of (8x^3+3)/((x^2+2)^2) is

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integral of (8x^3+3)/((x^2+2)^2)

\int \frac{8 x ^{3} + 3}{( x ^{2} + 2 ) ^{2}} d x

2 (22 ln \frac{x ^{2}}{2} + 1 + \frac{1}{4} (\frac{16 2}{x ^{2} + 2} + \frac{3}{2} (arctan (\frac{x}{2}) + \frac{2 x}{2 + x ^{2}}))) + C

Solution steps

\int \frac{8 x ^{3} + 3}{( x ^{2} + 2 ) ^{2}} d x

Apply Integral Substitution

= \int \frac{2 ( 16 2 u ^{3} + 3 )}{( 2 u ^{2} + 2 ) ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{16 2 u ^{3} + 3}{( 2 u ^{2} + 2 ) ^{2}} d u

Take the partial fraction of

\frac{16 2 u ^{3} + 3}{( 2 u ^{2} + 2 ) ^{2}} : \frac{4 2 u}{u ^{2} + 1} + \frac{- 16 2 u + 3}{4 ( u ^{2} + 1 ) ^{2}}

= 2 \cdot \int \frac{4 2 u}{u ^{2} + 1} + \frac{- 16 2 u + 3}{4 ( u ^{2} + 1 ) ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (\int \frac{4 2 u}{u ^{2} + 1} d u + \int \frac{- 16 2 u + 3}{4 ( u ^{2} + 1 ) ^{2}} d u)

\int \frac{4 2 u}{u ^{2} + 1} d u = 22 ln u^{2} + 1

\int \frac{- 16 2 u + 3}{4 ( u ^{2} + 1 ) ^{2}} d u = \frac{1}{4} (\frac{8 2}{u ^{2} + 1} + \frac{3}{2} (arctan (u) + \frac{u}{1 + u ^{2}}))

= 2 (22 ln u^{2} + 1 + \frac{1}{4} (\frac{8 2}{u ^{2} + 1} + \frac{3}{2} (arctan (u) + \frac{u}{1 + u ^{2}})))

Substitute back

u = \frac{x}{2}

= 2 22 ln (\frac{x}{2})^{2} + 1 + \frac{1}{4} \frac{8 2}{( \frac{x}{2} ) ^{2} + 1} + \frac{3}{2} arctan (\frac{x}{2}) + \frac{\frac{x}{2}}{1 + ( \frac{x}{2} ) ^{2}}

Simplify

2 22 ln (\frac{x}{2})^{2} + 1 + \frac{1}{4} \frac{8 2}{( \frac{x}{2} ) ^{2} + 1} + \frac{3}{2} arctan (\frac{x}{2}) + \frac{\frac{x}{2}}{1 + ( \frac{x}{2} ) ^{2}} : 2 (22 ln \frac{x ^{2}}{2} + 1 + \frac{1}{4} (\frac{16 2}{x ^{2} + 2} + \frac{3}{2} (arctan (\frac{x}{2}) + \frac{2 x}{2 + x ^{2}})))

= 2 (22 ln \frac{x ^{2}}{2} + 1 + \frac{1}{4} (\frac{16 2}{x ^{2} + 2} + \frac{3}{2} (arctan (\frac{x}{2}) + \frac{2 x}{2 + x ^{2}})))

Add a constant to the solution

= 2 (22 ln \frac{x ^{2}}{2} + 1 + \frac{1}{4} (\frac{16 2}{x ^{2} + 2} + \frac{3}{2} (arctan (\frac{x}{2}) + \frac{2 x}{2 + x ^{2}}))) + C

\frac{\partial}{\partial x} (sin^{2} (m x + n y))

\int 2^{s i n^{2} (\frac{1}{3} x)} sin (\frac{2}{3} x) d x

\frac{1}{x ^{2}}, - 1

\int (\frac{6 - 2 x}{8 - 4 x - 4 x ^{2}}) d x

\frac{d}{d x} (\frac{7}{x ^{4} + 3})

What is the integral of (8x^3+3)/((x^2+2)^2) ?
The integral of (8x^3+3)/((x^2+2)^2) is