What is the integral of (x^2+2)/(4x^5+4x^3+x) ?

The integral of (x^2+2)/(4x^5+4x^3+x) is 2ln|x|-ln|2x^2+1|+3/(4(2x^2+1))+C

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integral of (x^2+2)/(4x^5+4x^3+x)

\int \frac{x ^{2} + 2}{4 x ^{5} + 4 x ^{3} + x} d x

2 ln ∣ x ∣ - ln 2 x^{2} + 1 + \frac{3}{4 ( 2 x ^{2} + 1 )} + C

Solution steps

\int \frac{x ^{2} + 2}{4 x ^{5} + 4 x ^{3} + x} d x

Take the partial fraction of

\frac{x ^{2} + 2}{4 x ^{5} + 4 x ^{3} + x} : \frac{2}{x} - \frac{4 x}{2 x ^{2} + 1} - \frac{3 x}{( 2 x ^{2} + 1 ) ^{2}}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

= 2 ln ∣ x ∣ - ln 2 x^{2} + 1 - (- \frac{3}{4 ( 2 x ^{2} + 1 )})

Simplify

= 2 ln ∣ x ∣ - ln 2 x^{2} + 1 + \frac{3}{4 ( 2 x ^{2} + 1 )}

Add a constant to the solution

= 2 ln ∣ x ∣ - ln 2 x^{2} + 1 + \frac{3}{4 ( 2 x ^{2} + 1 )} + C

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What is the integral of (x^2+2)/(4x^5+4x^3+x) ?
The integral of (x^2+2)/(4x^5+4x^3+x) is 2ln|x|-ln|2x^2+1|+3/(4(2x^2+1))+C