Question

这道题答案里的不定积分是怎么算的

Answer 1

分部积分 ∫e^xsinxdx =∫sinxde^x =sinx*e^x-∫e^xdsinx =sinx*e^x-∫e^xcosxdx =sinx*e^x-∫cosxde^x =sinx*e^x-cosx*e^x+∫e^xdcosx =sinx*e^x-cosx*e^x-∫e^xsinxdx 移项所以2∫e^xsinxdx=sinx*e^x-cosx*e^x 所以∫e^xsinxdx=e^x(sinx-cosx)/2 再把上下限(0→π/2)代入 =e^(π/2)/2 - (0-1)/2 =(1/2)[e^(π/2) +1]

Answer 2

y = {∫cosxe^[∫(sinx/cosx)dx] dx + C} e^[-∫(sinx/cosx)dx]
= {∫cosxe^[∫(-1/cosx)dcosx] dx + C} e^[∫(1/cosx)dcosx]
= [∫cosxe^(-lncosx)dx + C] e^(lncosx)
= [∫cosxdx/cosx + C]cosx = (x+C)cosx