Integrate the function sin 3x*cos5x.

To calculate the indefinite integral of f(x)=sin3x*cos5x,
we'll  transform the product of trigonometric functions into a
sum.

We'll use the
formula:

sin a * cos b =
[sin(a+b)+sin(a-b)]/2

We'll substitute a by 3x and b by
5x.

sin3x*cos5x =
[sin(3x+5x)+sin(3x-5x)]/2

sin3x*cos5x = (sin 8x)/2 - (sin
2x)/2

Now, we'll calculate Int
f(x)dx.

Int sin3x*cos5x dx = Int (sin 8x)dx/2 - Int (sin
2x)dx/2

Int (sin 8x)dx = -(cos8x)/8 +
C

Int (sin2x)dx = -(cos 2x)/2 +
C

Int sin3x*cos5x dx = -(cos8x)/16 + (cos
2x)/4 + C