How do we derive
∫f(x) g(x) dx = f(x) G(x) - ∫ f'(x)G(x)dx
G(x) represents integral of g(x) wrt x
and f'(x) represents differential of f(x) wrt x
-
UP 0 DOWN 0 0 1
1 Answers
24consider f(x).g(x)
now d/dx f(x).g(x)=f(x).g'(x)+g(x).f'(x)
=> d(f(x).g(x))=f(x).g'(x) dx +g(x).f'(x) dx
=> ∫d(f(x).g(x))=∫f(x).g'(x) dx +∫g(x).f'(x) dx
=> f(x).g(x)=∫f(x).g'(x).dx +∫g(x).f'(x).dx --------------(1)
Let f(x)=u
g'(x)=v
=> d(g(x))= v.dx
=> ∫d(g(x))=∫ v.dx
=> g(x)=∫v.dx
put values in (1)
=> u∫v.dx=∫u.v.dx+∫(u'.∫v.dx).dx
=> ∫u.v.dx=u∫v.dx -∫(u'.∫v.dx).dx[1]
