1. ## 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]

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

2. Make

$\int \sin(3x)\cos(5x)~dx$

into

$\int \sin(4x-x)\cos(4x+x)~dx$

and using Sum-Difference formulas from here

Table of Trigonometric Identities

$\int \sin(4x-x)\cos(4x+x)~dx=$ $\int (\sin(4x)\sin(x)-\cos(4x)\cos(x))(\cos(4x)\cos(x)-\sin(4x)\sin(x))~dx$

now simplify and integrate.

3. Originally Posted by pickslides
Make

$\int \sin(3x)\cos(5x)~dx$

into

$\int \sin(4x-x)\cos(4x+x)~dx$

and using Sum-Difference formulas from here

Table of Trigonometric Identities

$\int \sin(4x-x)\cos(4x+x)~dx=$ $\int (\sin(4x)\sin(x)-\cos(4x)\cos(x))(\cos(4x)\cos(x)-\sin(4x)\sin(x))~dx$

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

4. $\sin(4x-x)=\sin(4x)\cos(x)-\cos(4x)\sin(x)$

5. 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]

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 $I=\int\sin3x\cos5x\,dx$

$u=\sin3x$ and $dv=\cos5x\,dx$
$du=3\cos3x\,dx$ and $v=\frac{1}{5}\sin5x$

The integration by parts formula gives:

$I=\frac{1}{5}\sin3x\sin5x-\frac{3}{5}\int\cos3x\sin5x\,dx$

$u=\cos3x$ and $dv=\sin5x\,dx$
$du=-3\sin3x\,dx$ and $v=-\frac{1}{5}\cos5x$

Now we have:

$I=\frac{1}{5}\sin3x\sin5x-\frac{3}{5}\int\cos3x\sin5x\,dx$ $=\frac{1}{5}\sin3x\sin5x-\frac{3}{5}\left[-\frac{1}{5}\cos3x\cos5x-\frac{3}{5}\int\sin3x\cos5x\,dx\right]$

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

$I=\frac{1}{5}\sin3x\sin5x-\frac{3}{5}\left[-\frac{1}{5}\cos3x\cos5x-\frac{3}{5}I\right]$

Solve for $I$. (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.