第2个回答 2022-02-16
∫arctane^xdx/e^(2x) = ∫e^x arctane^xdx/e^(3x)
= ∫arctane^xde^x/e^(3x) u = e^x
= ∫arctanudu/u^3 = (-1/2)∫arctanudu^(-2)
= (-1/2){u^(-2)arctanu -∫du/[u^2(1+u^2)]}
= (-1/2){u^(-2)arctanu -∫[1/u^2-1/(1+u^2)]du}
= (-1/2){arctanu/u^2 +1/u + arctanu} + C
= (-1/2)[arctan(e^u)/e^(2x) +1/e^x + arctan(e^x)] + C