What is the integral of 1/(x^2(2+x^2)) ?

The integral of 1/(x^2(2+x^2)) is

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integral of 1/(x^2(2+x^2))

\int \frac{1}{x ^{2} ( 2 + x ^{2} )} d x

\frac{- 2 x ^{5} arctan ( \frac{x}{2} ) - 2 \cdot 2 ^{- \frac{1}{2} + 1} x ^{4} - 4 2 x ^{3} arctan ( \frac{x}{2} ) - 8 x ^{2} - 4 2 x arctan ( \frac{x}{2} ) - 8}{4 x ( x ^{2} + 2 ) ^{2}} + C

Solution steps

\int \frac{1}{x ^{2} ( 2 + x ^{2} )} d x

Apply Integral Substitution

= \int \frac{1}{2 u ^{2} ( 2 + 2 u ^{2} )} d u

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{2} ( 2 + 2 u ^{2} )} d u

Apply Integration By Parts

: \frac{1}{u ( 2 u ^{2} + 2 )} - \int - \frac{2 u ^{2} + 1}{u ^{2} ( u ^{2} + 1 ) ^{2}} d u

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

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

= \frac{1}{2} (\frac{1}{u ( 2 u ^{2} + 2 )} - (\frac{u}{2 ( u ^{2} + 1 )} + \frac{1}{2} arctan (u) - \frac{u}{u ^{2} + 1} + \frac{1}{u}))

Substitute back

u = \frac{x}{2}

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

Simplify

\frac{1}{2} \frac{1}{\frac{x}{2} ( 2 ( \frac{x}{2} ) ^{2} + 2 )} - \frac{\frac{x}{2}}{2 ( ( \frac{x}{2} ) ^{2} + 1 )} + \frac{1}{2} arctan (\frac{x}{2}) - \frac{\frac{x}{2}}{( \frac{x}{2} ) ^{2} + 1} + \frac{1}{\frac{x}{2}} : \frac{- 2 x ^{5} arctan ( \frac{x}{2} ) - 2 \cdot 2 ^{- \frac{1}{2} + 1} x ^{4} - 4 2 x ^{3} arctan ( \frac{x}{2} ) - 8 x ^{2} - 4 2 x arctan ( \frac{x}{2} ) - 8}{4 x ( x ^{2} + 2 ) ^{2}}

= \frac{- 2 x ^{5} arctan ( \frac{x}{2} ) - 2 \cdot 2 ^{- \frac{1}{2} + 1} x ^{4} - 4 2 x ^{3} arctan ( \frac{x}{2} ) - 8 x ^{2} - 4 2 x arctan ( \frac{x}{2} ) - 8}{4 x ( x ^{2} + 2 ) ^{2}}

Add a constant to the solution

= \frac{- 2 x ^{5} arctan ( \frac{x}{2} ) - 2 \cdot 2 ^{- \frac{1}{2} + 1} x ^{4} - 4 2 x ^{3} arctan ( \frac{x}{2} ) - 8 x ^{2} - 4 2 x arctan ( \frac{x}{2} ) - 8}{4 x ( x ^{2} + 2 ) ^{2}} + C

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