分部积分
∫(0→π) (f(x)+f''(x))sinxdx
=∫(0→π) f(x)sinxdx+∫(0→π)f''(x)sinxdx
=∫(0→π) f(x)sinxdx+∫(0→π)sinxd(f'(x))
=∫(0→π) f(x)sinxdx+f'(x)sinx|(0→π)-∫(0→π)f'(x)d(sinx).注意：第二个式子等于0
=∫(0→π) f(x)sinxdx-∫(0→π)f'(x)cosxdx
=∫(0→π) f(x)sinxdx-∫(0→π)cosxd(f(x))
=∫(0→π) f(x)sinxdx-[f(x)cosx|(0→π)-∫(0→π)f(x)d(cosx)]
=∫(0→π) f(x)sinxdx-[f(x)cosx|(0→π)+∫(0→π)f(x)sinxdx]
=∫(0→π) f(x)sinxdx-f(x)cosx|(0→π)-∫(0→π)f(x)sinxdx
=-f(x)cosx|(0→π)