Question

大一高等数学习题求解大神在吗

打勾勾的三道小题
理工/定积分/高数/大一

Answer 1

(2)
let
u=√x
2udu =dx
x=1, u=1
x=4, u=2
∫(1->4) dx/(1+√x)
=∫(1->2) 2udu/(1+u)
=2∫(1->2) [1-1/(1+u)] du
=2[ u-ln|1+u|]|(1->2)
=2[ ( 2-ln3)-(1-ln2) ]
=2( 1 +ln2 -ln3 )
(4)
let
x=√2sinu
dx=√2cosu du
x=0, u=0
x=√2, u=π/2
∫(0->√2) √(2-x^2) dx
=2∫(0->π/2) (cosu)^2 du
=∫(0->π/2) (1+cos2u) du
=[u+(1/2)sin2u]|(0->π/2)
=π/2
(6)
let
u=e^x
du=e^x dx
x=ln2, u=2
x=ln3, u=3
∫(ln2->ln3) dx/[e^x -e^(-x) ]
=∫(2->3) (du/u)/(u -1/u )
=∫(2->3) du/(u^2 -1)
=(1/2)∫(2->3) [ 1/(u -1) -1/(u+1) ] du
=(1/2)[ ln|(u-1)/(u+1)|]|(2->3)
=(1/2)[ ln(2/4)- ln(1/3) ]
=(1/2)(ln3 -ln2)