Thread: Problems with a few Integrals

I just started Calc II and the first thing the teacher did was give us a review sheet of about 60 integrals and derivatives. There are 3 integrals in particular that I have been going around in circles with.

$=Integral

1) $cos^2[(1/2)x]dx

I know that the answer wanted is (1/2)x + (1/2)sinx + C and that the half angle formula gets me on on the correct path but I am missing something...

2) $sin^2(3x)dx

Again, I have been given the answer (1/2)x - (1/12)sin(6x) + C. This gives a hint to use the double angle forumua but I'll be darned if I can make it work.

3) $sin^4(x)dx

The answer given is (1/32)[12x-8sin(2x) + sin(4x)] + C. I have tried breaking this into (sin^2(x))^2 = (1 - cos^2(x))^2 = 1 - 2cos^2(x) + cos^4(x) and again I am doing something wrong.

I am sorry for posting 3 at a time. Any help/hints will be greatly appreciated!

Thank you in advance,

Quasar

Originally Posted by Quasar

1) $cos^2[(1/2)x]dx

Use half-angle identity,

The rest is simple,

2) $sin^2(3x)dx

Again half-angle identity,

Now,

And,

If you use .
Thus,

Originally Posted by Quasar

3) $sin^4(x)dx

You have,


Use half-angle identity,

Open sesame,

Use half-angle identity again,

Open sesame,

Combine the brethren,

Thus,

Factor,