integrate using the substution method xe^x^2 sin (e^x^2) cos (e^x^2) dx?

Answer 1

Let u == e^x^2 and du == 2 e^x^2 x dx, then substituting

1/2 Int_ sin[u] cos[u]] du
1/2 Int_ sin[u] d sin[u]
Substituting x == sin[u],
1/2 Int_ x dx
Solving,
1/4 x^2
Substituting back for x
1/4 sin[u]^2

Substituting back for u

ANS: 1/4 sin [ e^x^2 ] ^2

One can also do the substitution the other way:

-1/2 Int_ cos[u] d cos[u]
Substituting x == cos[u],
-1/2 Int_ x dx
Solving,
-1/4 x^2
Substituting back for x
-1/4 cos[u]^2

Substituting back for u

ANS: -1/4 cos [ e^x^2 ] ^2

Answer 2

When I figure out how to do the substitution (sorry, I can get the answer faster by thinking about it than writing it out), I'll post that, but for now the answer is sin^2(e^x^2)/4 or if you can't understand that, it's one fourth times sine squared of e^x^2

Answer 3

u=sin(e^(x^2)). Don't forget +C!