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

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

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

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

\frac{1}{2} ln x^{2} + 2 + ln (x) + C

Solution steps

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

Expand

\frac{2 x ^{3} + 2 x}{x ^{4} + 2 x ^{2}} : \frac{2 x ^{3}}{x ^{4} + 2 x ^{2}} + \frac{2 x}{x ^{4} + 2 x ^{2}}

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

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

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

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

Add similar elements:

ln x^{2} + 2 - \frac{1}{2} ln x^{2} + 2 = \frac{1}{2} ln x^{2} + 2

= \frac{1}{2} ln x^{2} + 2 + ln (x)

Add a constant to the solution

= \frac{1}{2} ln x^{2} + 2 + ln (x) + C

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