integration by parts

How do we derive

âˆ«f(x) g(x) dx = f(x) G(x) - âˆ« f'(x)G(x)dx

G(x) represents integral of g(x) wrt x
and f'(x) represents differential of f(x) wrt x

1 Answers

consider 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

=> âˆ«d(f(x).g(x))=âˆ«f(x).g'(x) dx +âˆ«g(x).f'(x) dx

=> f(x).g(x)=âˆ«f(x).g'(x).dx +âˆ«g(x).f'(x).dx --------------(1)
Let f(x)=u
g'(x)=v

=> d(g(x))= v.dx
=> âˆ«d(g(x))=âˆ« v.dx
=> g(x)=âˆ«v.dx

put values in (1)
=> uâˆ«v.dx=âˆ«u.v.dx+âˆ«(u'.âˆ«v.dx).dx
=> âˆ«u.v.dx=uâˆ«v.dx -âˆ«(u'.âˆ«v.dx).dx[1]

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