Evaluate `int(3x)/((1+2x^(4)))dx`.

1.	Evaluate `int(3x)/((1+2x^(4)))dx`.
Answer» Putting `x^(2)=t " and "2xdx=dt`, we get `int(3x)/((1+2x^(4)))dx=(3)/(2)int(dt)/((1+2t^(2)))` `=(3)/(4)int(dt)/(((1)/(2)+t^(2)))=(3)/(4)int(dt)/({((1)/(sqrt(2)))^(2)+t^(2)})` `=(3)/(4)(1)/((1//sqrt(2)))"tan"^(-1)(t)/((1//sqrt(2))^(2))+C` `=(3)/(2sqrt(2))tan^(-1)(sqrt(2)t)+C=(3)/(2sqrt(2))tan^(-1)(sqrt(2x)^(2))+C`.

Discussion

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