Help on these CALC questions please!? its on integrals...?

Question

I need help with these four problems.

(1)Evaluate the indefinite integral:
∫ e^(9x) sin (e^(9x)) dx

The wrong answer i got for this question was:
-cos(e^(9x)) + C

(2)Evaluate the integral by making the given subsitution:
∫ ((sin(√x)) / (√x)) dx , u=√x

The wrong answer I got for this one is:
sin(e^(√x)) +C

(3)Evaluate the definite integral:
∫1 to 2 ((e^(1/x^5))/(x^6)) dx

The answer I got for this one:
((e^(1/32))/-16) - ((e^(1))/-4)

(4)Evaluate the definite integral:
∫ (-π/3) to (π/3) ((x^(2)sin(x))/(1+x^(2))) dx

I wasn't able to even get close to an answer for this one.



I'd appreciate the help. Thank you.

Answer 1

For the first one, let u = e^(9x), so du = 9e^(9x):

∫ e^(9x) sin (e^(9x)) dx = (1/9) ∫ sin u du
= (-cos u)/9 + C
= (-cos (e^(9x)) /9 + C

For the second one, if u = √x, then du = 1/√x (if u = x^(1/2), then du = x^(-1/2)):

∫ ((sin(√x)) / (√x)) dx = ∫ sin u du
= -cos u + C
= -cos(√x)

For the third one, if u = x^(-5), then du = x^(-6):

∫1 to 2 ((e^(1/x^5))/(x^6)) dx = ∫ e^u du
= e^u
= e^(x^(-5)) from 1 to 2
= e^(2^(-5)) - e^(1^(-5))
= e^(1/32) - e^(1/5)

I'm not sure on the last one either. I've tried integration by parts and by u substitution, but those don't seem to work. There's probably some obscure integral that relates the squared terms to the sin term, but it's escaping me right now.