Question

大一高数不定积分相关问题

如图，这两个有什么区别，请分别解释一下代表什么意思

Answer 1

一个是求导数，一个是求微分

Answer 2

∫cos(√x)dx
令√x=u,则dx/2√x=du,dx=2(√x)du=2udu,
原式=2∫ucosudu
=2∫ud(sinu)
=2[usinu-∫sinudu]
=2(usinu+cosu)+C
=2[(√x)sin(√x)+cos(√x)]+C
~~~~~~~~~~~~~~~~~~~~~~~~~
∫√x(x+1)^2dx
令√x=t, 则dx=2tdt，带入
=∫t(t^2+1)^2*2tdt
=∫2t^6+4t^4+2t^2dt
=2/7t^7+4/5t^5+2/3t^3+c
反带回
=2/7(√x)^7+4/5(√x)^5+2/3(√x)^3+c
~~~~~~~~~~~~
∫e^x/(1+e^x)^(1/2)dx
=∫2d[(1+e^x)^(1/2)]
=2(1+e^x)^(1/2)+c

Answer 3

d/dx ∫ f(x) dx = f(x)
d∫ f(x) dx = f(x) dx