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

The integral of ((x^2))/(((2x+x^3)^3)) is 1/8 ln|x|-1/16 ln|x^2+2|+1/(8(x^2+2))+1/(8(x^2+2)^2)+C

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

integral

\frac{( x ^{2} )}{( ( 2 x + x ^{3} ) ^{3} )}

\frac{1}{8} ln ∣ x ∣ - \frac{1}{16} ln x^{2} + 2 + \frac{1}{8 ( x ^{2} + 2 )} + \frac{1}{8 ( x ^{2} + 2 ) ^{2}} + C

Solution steps

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

Take the partial fraction of

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

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

\int \frac{1}{8 x} d x = \frac{1}{8} ln ∣ x ∣

\int \frac{x}{8 ( x ^{2} + 2 )} d x = \frac{1}{16} ln x^{2} + 2

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

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

= \frac{1}{8} ln ∣ x ∣ - \frac{1}{16} ln x^{2} + 2 - (- \frac{1}{8 ( x ^{2} + 2 )}) - (- \frac{1}{8 ( x ^{2} + 2 ) ^{2}})

Simplify

= \frac{1}{8} ln ∣ x ∣ - \frac{1}{16} ln x^{2} + 2 + \frac{1}{8 ( x ^{2} + 2 )} + \frac{1}{8 ( x ^{2} + 2 ) ^{2}}

Add a constant to the solution

= \frac{1}{8} ln ∣ x ∣ - \frac{1}{16} ln x^{2} + 2 + \frac{1}{8 ( x ^{2} + 2 )} + \frac{1}{8 ( x ^{2} + 2 ) ^{2}} + C

What is the integral of ((x^2))/(((2x+x^3)^3)) ?
The integral of ((x^2))/(((2x+x^3)^3)) is 1/8 ln|x|-1/16 ln|x^2+2|+1/(8(x^2+2))+1/(8(x^2+2)^2)+C