求∫arctan（e^x）/（e^x）dx?

题目

求∫arctan（e^x）/（e^x）dx?

答案

a=e^x
x=lna
dx=da/a
所以原式=∫arctana*da/a²
=-∫arctanad(1/a)
=-arctana/a+∫1/a*darctana
=-arctana/a+∫1/a*da/(1+a²)
∫1/a*da/(1+a²)
=∫(1+a²-a²)/a(a²+1)da
=∫[1/a-a/(a²+1)]da
=∫1/ada-∫a/(a²+1)da
=lna-1/2∫d(a²+1)/(a²+1)
=lna-1/2*ln(a²+1)+C
所以原式=-arctana/a+lna-1/2*ln(a²+1)+C
=-arctan(e^x)/e^x+x-1/2*ln(e^2x+1)+C