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

The integral of 4/((x^2+2^2)^2) is

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

\int \frac{4}{( x ^{2} + 2 ^{2} ) ^{2}} d x

4 (\frac{x}{( x ^{2} + 4 ) ^{2}} + 4 (\frac{1}{16} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) - \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{64 ( x ^{2} + 4 ) ^{2}})) + C

Solution steps

\int \frac{4}{( x ^{2} + 2 ^{2} ) ^{2}} d x

Take the constant out:

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

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

Apply Integration By Parts

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

\int - \frac{4 x ^{2}}{( x ^{2} + 4 ) ^{3}} d x = - 4 (\frac{1}{16} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) - \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{64 ( x ^{2} + 4 ) ^{2}})

= 4 (\frac{x}{( x ^{2} + 4 ) ^{2}} - (- 4 (\frac{1}{16} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) - \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{64 ( x ^{2} + 4 ) ^{2}})))

Simplify

= 4 (\frac{x}{( x ^{2} + 4 ) ^{2}} + 4 (\frac{1}{16} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) - \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{64 ( x ^{2} + 4 ) ^{2}}))

Add a constant to the solution

= 4 (\frac{x}{( x ^{2} + 4 ) ^{2}} + 4 (\frac{1}{16} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) - \frac{3 x ^{4} arctan ( \frac{x}{2} ) + 6 x ^{3} + 24 x ^{2} arctan ( \frac{x}{2} ) + 40 x + 48 arctan ( \frac{x}{2} )}{64 ( x ^{2} + 4 ) ^{2}})) + C