Which is the indefinite integral of f(x)=sin5x*cos3x ?

To calculate the indefinite integral of f(x)=sin5x*cos3x,
we'll apply the formula to 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 5x and b by
3x.

sin5x*cos3x =
[sin(5x+3x)+sin(5x-3x)]/2

sin5x*cos3x = sin 8x/2 +
sin2x/2

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

Int sin5x*cos3xdx = Int (sin 8x)dx/2 + Int
(sin2x)dx/2

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

Int (sin2x)dx = -(cos2x)/2 +
C

Int sin5x*cos3xdx = -(cos8x)/8 - (cos2x)/2
+ C