Question

计算下列定积分: ∫上限1下限0(xe^x)dx; ∫上限1e下限0xlnxdx；求过程！！！

Answer 1

∫(0→1) xe^x dx = ∫(0→1) x d(e^x)
= xe^x - ∫(0→1) e^x dx
= [(1)e^(1) - (0)e^(0)] - e^x
= e - [e^(1) - e^(0)]
= e - e + 1
= 1

∫(0→e) xlnx dx = ∫(0→e) lnx d(x²/2)
= (1/2)x²lnx - (1/2)∫(0→e) x² d(lnx)
= [(1/2)(e²)ln(e) - (1/2)(0)] - (1/2)∫(0→e) x dx
= (1/2)e² - (1/2)(x²/2)
= (1/2)e² - (1/4)(e² - 0)
= (1/4)e²

追问

第一个算对了！第二个我书上的答案是e²/4+1/4=1/4(1+e²)

追答

可能你上限打错了吧，1e什麽意思？

追问

好意思！上限是个e，没有1！不好意思啊！

追答

∫ xlnx dx = ∫ lnx d(x²/2) = (1/2)∫ lnx d(x²)
= (1/2)x²lnx - (1/2)∫ x² d(lnx)
= (1/2)x²lnx - (1/2)∫ x²(1/x) dx
= (1/2)x²lnx - (1/2)∫ x dx
= (1/2)x²lnx - (1/2)(x²/2) + C
= (1/2)x²lnx - (1/4)x² + C
= (1/4)x²(2lnx - 1) + C
代入上下限得：
= (1/4)(e²)[2ln(e) - 1] - (1/4)(0)
= (1/4)(e²)(2 - 1)
= e²/4
还有意见不？你要有鉴别能力，不只是光信答案就可以。

追问

谢谢了啊！一会给你发过去几道题！帮忙解答一下！