What is the antiderivative of f(x)*cos x if f(x)=ln(1+sin^2x)?

To determine the antiderivative of the product f(x)*cos x,
we'll have to determine the indefinite integral of f(x)*cos
x.

Int f(x)*cos x dx = Int ln[1+(sin x)^2]*cos x
dx

We'll replace sin x by
t:

sin x = t

We'll
differentiate both sides:

cos x dx =
dt

We'll re-write the
integral:

Int
ln[1+(t)^2]dt

We'll integrate by parts using the
formula:

Int udv = u*v - Int
vdu

u = ln(1+t^2) => du = 2t
dt/(1+t^2)

dv = dt => v =
t

Int ln[1+(t)^2]dt = t*ln(1+t^2) - Int 2t^2
dt/(1+t^2)

Int 2t^2 dt/(1+t^2) = 2 Int dt - 2 Int
dt/(1+t^2)

Int 2t^2 dt/(1+t^2) = 2t - 2 arctan t +
C

The antiderivative of the given function
is:

Int ln[1+(t)^2]dt = t*ln(1+t^2) - 2t + 2 arctan t +
C

Int ln[1+(sin x)^2]*cos x dx =sin
x*ln[1+(sin x)^2] - 2sin x + 2*arctan (sin x) + C