This integral

Tan(x)^3+Tan(x)

doing it by hang gives Tan(x)^2/2

Mathematica gives Sec(x)^2/2

 

so why the difference, are both correct, how to get the mathematica answer by hand?

Since sec^2(X) = tan^2(x) + 1, the only difference will be in the integration constant

I see, I hope the exam does not give mathematica answer as the chice because I have absolutely no clue how to get sec(x)^2/2, anyone know how to get that?

tartaglia already said....... tan²(x)+1=sec²(x)

ok, now I understand it, basically constant can be thrown out.

there's still a constant, it is just a different one.

 

you get .5tan²(x)+C. they get .5sec²(x)+C'. the answers are identicle. C and C' differ by .5.

one observation:

 

sec^2=tan^2+1

 

has been used to explain how to translate between the two answers, after the integration, but it can be used bbefore hand too. use it in the original expression and you just get

 

tan(x)sec^2(x)

 

to integrate, which is sin(x)/cos^3(x), and that is integrable by inspection as 1/2cos^2(x) = sec^2(x).