Thread: Intergration

This problem I cannot seem to solve,I have the answer but i want to know how to get to it, i do not know if i used the table or do i integrate it by parts? Thank You!!
The problem:
Integrate[Sin[3 x] Cos[5 x], x]

and this is the answer:

8 (Cos[x]) ^(2)- Cos[8 x]
-------------------------
16

Make



into



and using Sum-Difference formulas from here

Table of Trigonometric Identities



now simplify and integrate.

Originally Posted by pickslides

Make



into



and using Sum-Difference formulas from here

Table of Trigonometric Identities



now simplify and integrate.

at the end when you plug back in the trig identities, the sum-difference,from sin(4x-x) is not right

Originally Posted by flutterby

This problem I cannot seem to solve,I have the answer but i want to know how to get to it, i do not know if i used the table or do i integrate it by parts? Thank You!!
The problem:
Integrate[Sin[3 x] Cos[5 x], x]

and this is the answer:

8 (Cos[x]) ^(2)- Cos[8 x]
-------------------------
16

You can also do this using the "cyclic method". (That's not it's official name; that's just what I call it.)

Let

and
and

The integration by parts formula gives:



and
and

Now we have:



Note that that last integral is just , so we have:



Solve for . (You should also check my math; make sure I didn't make any sign errors and whatnot.)

Remark: To get the answer into the format given to you in the answer key, you may need to use some trig identities.