Question

How do you integrate 2sin5xsinx?

Answer 1

I did a search for "trig sum and difference identities" and came up with the following URL:

http://www.purplemath.com/modules/idents.htm

Within that page, look for the heading "Sum Identities."

Hope this gets you onto the path towards solving this problem.

Hint:

Suppose

a+b=5 and
a-b =1

Solve for a and b.

Answer 2

So 2a=6
a=3

3+b=5
b=2

Then what do I do next?

Answer 3

Also, does the formula under "Product Identities" work as well?

Answer 4

Actually, those formulas under Product identities are faster to use. Glad that you've noticed that.

Starting with 2sin5xsinx, I'd hold the factor, 2, and concentrate on the sin5x*sinx part.

Looks like the very last one, for sin x*sin y, is the one we want.

Answer 5

Care to try evaluating sin 5x * sin x using this formula?

When done, be certain to multiply the entire result by 2. (Why?)

Answer 6

So using sin(P)sin(Q)=1/2[ 2cos(x-y) - 2cos(x+y) ]
= 1/2 [ 2cos (5x-x) - 2cos (5x+x) ]
= 1/2 [ 2cos4x - 2cos6x]

Would the final answer be this -

= (1/2) (1/4) 2sin4x - (1/2) (1/6) 2sin6x
= 1/4 sin4x - 1/6 sin6x

???