Menu Close

f-and-g-are-2-function-C-n-on-a-b-prove-that-a-b-f-n-x-g-x-dx-k-0-n-1-1-k-f-k-g-n-k-a-b-1-n-a-b-f-x-g-n-x-dx-




Question Number 30564 by abdo imad last updated on 23/Feb/18
f and g are 2 function  C^n  on [a,b] prove that  ∫_a ^b  f^((n)) (x)g(x)dx=[Σ_(k=0) ^(n−1) (−1)^k  f^((k)) g^((n−k)) ]_a ^b  +(−1)^n ∫_a ^b f(x)g^((n)) (x)dx
$${f}\:{and}\:{g}\:{are}\:\mathrm{2}\:{function}\:\:{C}^{{n}} \:{on}\:\left[{a},{b}\right]\:{prove}\:{that} \\ $$$$\int_{{a}} ^{{b}} \:{f}^{\left({n}\right)} \left({x}\right){g}\left({x}\right){dx}=\left[\sum_{{k}=\mathrm{0}} ^{{n}−\mathrm{1}} \left(−\mathrm{1}\right)^{{k}} \:{f}^{\left({k}\right)} {g}^{\left({n}−{k}\right)} \right]_{{a}} ^{{b}} \:+\left(−\mathrm{1}\right)^{{n}} \int_{{a}} ^{{b}} {f}\left({x}\right){g}^{\left({n}\right)} \left({x}\right){dx} \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *