∫dx/(1+e^x)^2=?,x∈[0,ln2]错了错了改为∫e^x(1+e^x)^2dx=?x∈[0,ln2],请回答
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![∫dx/(1+e^x)^2=?,x∈[0,ln2]错了错了改为∫e^x(1+e^x)^2dx=?x∈[0,ln2],请回答](/uploads/image/z/3892085-53-5.jpg?t=%E2%88%ABdx%2F%281%2Be%5Ex%29%5E2%3D%3F%2Cx%E2%88%88%5B0%2Cln2%5D%E9%94%99%E4%BA%86%E9%94%99%E4%BA%86%E6%94%B9%E4%B8%BA%E2%88%ABe%5Ex%281%2Be%5Ex%29%5E2dx%3D%3Fx%E2%88%88%5B0%2Cln2%5D%EF%BC%8C%E8%AF%B7%E5%9B%9E%E7%AD%94)
∫dx/(1+e^x)^2=?,x∈[0,ln2]错了错了改为∫e^x(1+e^x)^2dx=?x∈[0,ln2],请回答
∫dx/(1+e^x)^2=?,x∈[0,ln2]
错了错了
改为∫e^x(1+e^x)^2dx=?x∈[0,ln2],请回答
∫dx/(1+e^x)^2=?,x∈[0,ln2]错了错了改为∫e^x(1+e^x)^2dx=?x∈[0,ln2],请回答
见图
∫dx/(1+e^x)^2
= ∫(1+e^x)/(1+e^x)^2dx -∫e^x/(1+e^x)^2dx
=∫1/(1+e^x)dx -∫e^x/(1+e^x)^2dx
=∫(1+e^x)/(1+e^x)dx-∫e^x/(1+e^x)dx -∫e^x/(1+e^x)^2dx
=∫1dx-∫e^x/(1+e^x)dx -∫e^x/(1+e^x)^2dx
全部展开
∫dx/(1+e^x)^2
= ∫(1+e^x)/(1+e^x)^2dx -∫e^x/(1+e^x)^2dx
=∫1/(1+e^x)dx -∫e^x/(1+e^x)^2dx
=∫(1+e^x)/(1+e^x)dx-∫e^x/(1+e^x)dx -∫e^x/(1+e^x)^2dx
=∫1dx-∫e^x/(1+e^x)dx -∫e^x/(1+e^x)^2dx
=x-ln(1+e^x)+1/(1+e^x)+C
带入积分区间[0,ln2]
=ln2-ln3+1/3
=1/3+ln(2/3)
答案有误吧
收起
∫e^x(1+e^x)^2dx=
u=e^x
∫u(u+1)^2dx
=∫u^3dx+2∫u^2dx+∫udx
=∫u^2du+2∫udu+∫du
=u^3/3+u^2+u
=e^3x/3+e^2x+e^x+c
f(x)=e^3x/3+e^2x+e^x+c
∫e^x(1+e^x)^2dx=f(ln2)-f(0)
=8/3+4+2-1/3-1-1
=19/3