Find f(x)?

Question

If ∫ f(x)sinx dx=-f(x)cosx+∫ 3x^2*cosx dx, find f(x).

 

Answer: x^3

 

Please show all your work.

Answer 1

A method that's equivalent to Kirill's integration by parts is to take the derivative of this integral equation, turning it into a differential equation:

 

∫ f(x) sinx dx = -f(x)cosx + ∫ 3x² cosx dx

 

f(x) sinx = -f'(x)cosx + f(x)sinx + 3x² cosx

 

0 = -f'(x)cosx + 3x² cosx

 

f'(x) = 3x²

 

f(x) = x³ + C.

Answer 2

∫fd(g)=fg-∫gd(f)  -- integration by parts. 

 

In your example:

 

∫f(x)sinx dx=∫f(x) d(-cosx)=-f(x)cosx-∫(-cosx)df(x)=-f(x)cosx+∫cosx*df(x)=-f(x)cosx+∫cosx*f'(x)dx

 

By comparison, f'(x)=3x². Thus, f(x)=x³+C. If we take C=0, then

 

Answer: f(x)=x³