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

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

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integral of 4/((1+2x^2)^3)

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

\frac{4 x}{( 1 + 2 x ^{2} ) ^{3}} + \frac{6 x}{( 2 x ^{2} + 1 ) ^{2}} + \frac{9 x}{2 x ^{2} + 1} + \frac{9}{2} arctan (2 x) - \frac{4 x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} + \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{4 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1} + C

Solution steps

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

Apply Integration By Parts

: \frac{x}{( 1 + 2 x ^{2} ) ^{3}} - \int - \frac{12 x ^{2}}{( 1 + 2 x ^{2} ) ^{4}} d x

= 4 (\frac{x}{( 1 + 2 x ^{2} ) ^{3}} - \int - \frac{12 x ^{2}}{( 1 + 2 x ^{2} ) ^{4}} d x)

\int - \frac{12 x ^{2}}{( 1 + 2 x ^{2} ) ^{4}} d x = - \frac{3 x}{2 ( 2 x ^{2} + 1 ) ^{2}} - \frac{9 x}{4 ( 2 x ^{2} + 1 )} - \frac{9}{4 2} arctan (2 x) + \frac{x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} - \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{16 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1}

= 4 \frac{x}{( 1 + 2 x ^{2} ) ^{3}} - - \frac{3 x}{2 ( 2 x ^{2} + 1 ) ^{2}} - \frac{9 x}{4 ( 2 x ^{2} + 1 )} - \frac{9}{4 2} arctan (2 x) + \frac{x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} - \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{16 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1}

Simplify

4 \frac{x}{( 1 + 2 x ^{2} ) ^{3}} - - \frac{3 x}{2 ( 2 x ^{2} + 1 ) ^{2}} - \frac{9 x}{4 ( 2 x ^{2} + 1 )} - \frac{9}{4 2} arctan (2 x) + \frac{x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} - \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{16 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1} : \frac{4 x}{( 1 + 2 x ^{2} ) ^{3}} + \frac{6 x}{( 2 x ^{2} + 1 ) ^{2}} + \frac{9 x}{2 x ^{2} + 1} + \frac{9}{2} arctan (2 x) - \frac{4 x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} + \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{4 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1}

= \frac{4 x}{( 1 + 2 x ^{2} ) ^{3}} + \frac{6 x}{( 2 x ^{2} + 1 ) ^{2}} + \frac{9 x}{2 x ^{2} + 1} + \frac{9}{2} arctan (2 x) - \frac{4 x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} + \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{4 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1}

Add a constant to the solution

= \frac{4 x}{( 1 + 2 x ^{2} ) ^{3}} + \frac{6 x}{( 2 x ^{2} + 1 ) ^{2}} + \frac{9 x}{2 x ^{2} + 1} + \frac{9}{2} arctan (2 x) - \frac{4 x}{( 2 x ^{2} + 1 ) ^{\frac{5}{2}} 2 x ^{2} + 1} + \frac{5 ( - 4 2 x + 3 ( 2 arccot ( 2 x ) - sin ( 2 arccot ( 2 x ) ) ) ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1 )}{4 2 ( 2 x ^{2} + 1 ) ^{\frac{3}{2}} 2 x ^{2} + 1} + C

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