Anti derivative of cos(x)cos(sin(x))?

Question

Anti derivative of cos(x)cos(sin(x))?

Answer 1

the answer is sin(sinx) +c

by substitution: let u = sinx thus du = cosx dx

the integral becomes cosu du which is sin (u) +c

Answer 2

∫ cosx cos(sinx) dx =

note that your integrand includes both the function sinx (cosine argument)

and its derivative cosx, thus let sinx = u

differentiate both sides:

d(sinx) = du →

cosx dx = du

thus, susbtituting, you get:

∫ cos(sinx) cosx dx = ∫ cos u du = sin u + C

thus, substituting back u = sinx, you get:

∫ cos(sinx) cosx dx = sin(sinx) + C

I hope it helps...

Bye!