Question

高数:求不定积分

Answer 1

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Answer 2

(1)
let
u=e^x
du =e^x dx
dx = du/u
∫ dx/(1+e^x)
=∫ (du/u) /(1+u)
=∫ du/[u(1+u]
=∫ [1/u-1/(1+u] du
=ln|u| -ln|1+u| + C
=x -ln|1+e^x| +C
(2)
let
u=e^x
du/u =dx
∫ dx/[ e^(-x) +e^x]

=∫ (du/u) /[ 1/u +u]
=∫ du/(1+u^2)
=arctanu + C
=arctan(e^x) + C
(3)
∫(3x^2-2)/(x^3-2x+1) dx
=∫d(x^3-2x+1)/(x^3-2x+1)
=ln|x^3-2x+1| +C
(4)
∫[ e^(2x)+2e^(3x) +2] .e^x dx
=∫[ e^(3x)+2e^(4x) +2e^x ] dx
=(1/3)e^(3x)+(1/2)e^(4x) +2e^x + C

Answer 3