Integration by parts formula Theorem If u and vare two function of x then ∫(uv)dx =[u.∫vdx]-∫{du/dx.∫vdx}dx Proof For any two functions f₁(x)andf₂(x),we have d/dx[f₁(x).f₂(x)]=f₁(x).f₂'(x)+f₂(x).f₁'(x) ∫{f₁(x).f₂'(x)+f₂(x).f₁'(x)}dx=f₁(x).f₂(x) ∫{f₁(x).f₂'(x)}dx+∫{f₂(x).f₁'(x)}dx=f₁(x).f₂(x) ∫{f₁(x).f₂'(x)}dx=f₁(x).f₂(x)-∫{f₂(x).f₁'(x)}dx Let f₁(x)=u and f₂'(x)=v so that f₂(x)=∫vdx ∴∫(uv)dx=u.∫vdx-∫{du/dx.∫vdx}dx We can express this result as given below: Integral of product of two functions = (1st function ) X (integral […]
