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

The integral of (5x^2+3)/(x^2(x^2+2)) is -3/(2x)+7/(2sqrt(2))arctan(x/(sqrt(2)))+C

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

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

- \frac{3}{2 x} + \frac{7}{2 2} arctan (\frac{x}{2}) + C

Solution steps

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

Take the partial fraction of

\frac{5 x ^{2} + 3}{x ^{2} ( x ^{2} + 2 )} : \frac{3}{2 x ^{2}} + \frac{7}{2 ( x ^{2} + 2 )}

= \int \frac{3}{2 x ^{2}} + \frac{7}{2 ( x ^{2} + 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{3}{2 x ^{2}} d x + \int \frac{7}{2 ( x ^{2} + 2 )} d x

\int \frac{3}{2 x ^{2}} d x = - \frac{3}{2 x}

\int \frac{7}{2 ( x ^{2} + 2 )} d x = \frac{7}{2 2} arctan (\frac{x}{2})

= - \frac{3}{2 x} + \frac{7}{2 2} arctan (\frac{x}{2})

Add a constant to the solution

= - \frac{3}{2 x} + \frac{7}{2 2} arctan (\frac{x}{2}) + C

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What is the integral of (5x^2+3)/(x^2(x^2+2)) ?
The integral of (5x^2+3)/(x^2(x^2+2)) is -3/(2x)+7/(2sqrt(2))arctan(x/(sqrt(2)))+C