a) integral of sin2xdx/e^x, b) integral of sin(lny)dy, c)integral of x^5e^x^2dx, d) integral tan^-1sqrt(x)dx?

Answer 1

a) let u=sin 2x then u'=2cos 2x and v'=e^-x then v=-e^-x. using integration by parts
integral uv'=-e^-x*sin2x+2I[(cos 2x)/e^x]
2I[(cos 2x)/e^x]=-{(2cos 2x)/e^x +4I[(sin 2x)/e^x]
combining the equations we have
5integral of sin2xdx/e^x=-(sin 2x + 2cos 2x)/e^x
hence
integral sin2xdx/e^x=-[sin 2x +2cos 2x]/e^x +C
b). integral of sin(lny)dy let t=lny, e^tdt=dy and
integral of sin(lny)dy=I{e^tsin tdt} using by parts I got
integral of sin(lny)dy=(y/2)[sin(lny) -cos(lny)] +C

That's as far I can go

Answer 2

2+2=5

Answer 3

I'll do the last first, see how far I get:
let w = sqrt x then dw =1/2 x^(- 1/2)dx and
dx = 2wdw. So we want to integrate 2w arctan w which you do by parts picking u = arctan w and dv = 2w dw
so uv -vdu = w^2*arctan w - w^2dw/(1 + w^2)
We have to integrate that last term but
w^2/(1 + w^2) = 1 - 1/(1 + w^2) which we can integrate directly so with the first term above we have
w^2arctanw - w + arctanx + C
Phew!!
If you still need the others done let me know and I'll work on them. [e-mail]