How to solve single variable integrals with specific limits and constants?

[wam_mi]

Homework Statement

Hi could anyone please help me to solve the following integrals?
x and z are the only variable in all of the following integrals. Others are just constants.

(a) Integrate sin(2x) * [(sinx)^2] with limits [0,a] or with limits [-a, +a]

(b) Integrate sin(3x) * [(sinx)^2] with limits [0,a]

Does it require the same method for cos, for example cos(5x) * (cosx)^3

(c) Integrate the indefinite integral 1/((x^2)+4)
I guess this one has something to do with arctan, but I don't know how to start with it.
Could anyone please show me the steps?

Okay, I promise this is the last question...
(d) Integrate the indefinite integral exp(-kz)/(z+iL) where k is a positive constant, such that z=iL is the singularity inside any simple closed contour C(R).

I try to find the residue at z=-iL = (exp(-kz) / first derivative of x+iL and then evaluate everything at z=-iL which gives = exp(-ikL)

Then the required integral is 2*pi*exp(ikL).
But the question is, is that right or have I done anything fundamentally wrong?

Homework Equations

The Attempt at a Solution

Thanks a lot, that's it from me!

 

[jgens]

Few brief comments.

For a, I would recommend that you use the identity sin(2x) = 2sin(x)cos(x) and then make a proper choice for u

For b, I believe a similar method using the identity sin(3x) = 3cos^2(x)sin(x) - sin^3(x) or a variant of it would work.

For c, try the substitution 2u = x. Than x^2 = 4u^2. After that, integrate the resultant function.