Question

大一微积分，最好有详细步骤

，大一微积分，第四五六题最好有详细步骤

Answer 1

(4)三角换元脱根号，令x=√2sinu
=∫2sin²u/√2cosud√2sinu
=∫1-cos2udu
=u-sin2u/2+C
=arcsin(x/√2)-x√(2-x²)/2+C
(5)凑微分后换元u=x²，分部积分法
=1/2∫x^4e^x²dx²
=1/2∫u²de^u
=u²e^u/2-1/2∫e^udu²
=u²e^u/2-∫ude^u
=u²e^u/2-ue^u+e^u+C
代入u=x²还原
(6)换元脱根号，令u=√x
=∫sinudu²
=-2∫udcosu
=-2ucosu+2∫cosu
=-2√xcos√x+2sin√x+C

Answer 2

如上图所示。

追答

Answer 3

=∫(0到π/2)sinu/e^2ude^u
=∫sinue^-udu=-∫sinude^-u
=-sinue^-u+∫e^-udsinu
=-sinue^-u-∫cosude^-u
=-sinue^-u-cosue^-u+∫e^-udcosu
=-sinue^-u-cosue^-u-∫sinue^-udu
所以定积分=(-1/2)(sinue^-u+cosue^-u)