Question

大一数学定积分，大神来来来

Answer 1

f(0)=a, f(π)=b

∫(0->π) [f(x)+ f''(x)] sinx dx
=∫(0->π) f(x) sinx dx +∫(0->π) f''(x) sinx dx
=(a+b)+∫(0->π) cosx .f'(x) dx -∫(0->π) cosx .f'(x) dx
=a+b

consider
∫(0->π) f(x) sinx dx
=-∫(0->π) f(x) dcosx
=-[f(x).cosx]|(0->π) + ∫(0->π) cosx .f'(x) dx
=(a+b)+∫(0->π) cosx .f'(x) dx

∫(0->π) f''(x) sinx dx
=∫(0->π) sinx df'(x)
=[sinx. f'(x)]|(0->π) - ∫(0->π) cosx.f'(x) dx
=- ∫(0->π) cosx.f'(x) dx