Question

求定积分

能提供解题过程吗？我解的和标准答案不同！

Answer 1

设t=根号(e^x-1)
x=ln(t^2+1)
dx=2t/(t^2+1)
且x=0,t=0;x=ln2,t=1

以下省略积分号...因为不会打 囧
积分号 根号(e^x-1)dx
=积分号 t*2tdt/(t^2+1)
=积分号(2t^2+2)dt/(t^2+1)-积分号2dt/(t^2+1)
=积分号2dt-积分号2dt/(t^2+1)
=(2t-2arctant)|带入t上下限1和0
=2-派/2

Answer 2

Answer 3

设e^x-1=t^2,0=<t<=1
原式=t*2t/(1+t^2)dt求积分
=[2-2*1/(1+t^2)]dt
=2t-2arctant (t=0->1)
=2-2(π /4-0)=2-π/2

Answer 4