Question

Consider the integral ∫[xsec²x] dx.

If we try to apply Integration by Parts, we have (at least) two choices.

(1) First, we can do

u=x dv=sec2xdx implying

du= Answer v= Answer

Then

∫[xsec²x] dx = Answer - ∫tanx dx = xtanx + Answer + C

Hence, this choice of u and dv worked great.

(2) We could have tried

u=sec²x dv=xdx leading to

du=2sec²xAnswer v=(1/2)Answer

Then

∫[xsec²x] dx = (1/2)Answersec²x - ∫[x²sec²xtanx] dx,

and the last integral seems (is) more complicated than the original one, ∫[xsec2x] dx, hence, the second choice is not a good one.