5
令t=2x+1
∫(0到1)f(2x+1)dx=1/2∫(0到1)f(2x+1)d(2x+1)=1/2∫(1到3)f(t)dt=1/2
三大题
第一小题
1
=(x^3-x^2/2+x) | 0到a
=a^3-a^2/2+a
2
原式=arcsinx |(-1/2到1/2)
=π/3
3
=4∫(0到π/2)sinxdx=4
4
(3x^4+3x^2+1)/(x^2+1)
=(3x^2(x^2+1)+1)/(x^2+1)
=3x^2+ 1/(x^2+1)
所以积分=∫(-1到0)【3x^2+ 1/(x^2+1)】dx
=x^3+arctanx |(-1到0)
=1+π/4
5
∫(tanx)^2 dx=∫ [(secx)^2-1]dx= tanx-x |(0到π/4)
=1-π/4
6
令t=√x,x=t^2 dx=2tdt
原积分=2∫(2到3)t^2(1+t)dt=2t^3/3+t^2 | (2到3)
=53/3
第二小题
1
令t=x-π/3
=∫(0到π/6)costdt=1/2
2
1/(x^2+2x+2)=1/[1+(x+1)^2]
令t=x+1
原积分=∫(-2到0) 1/[1+(x+1)^2] d(x+1)
=∫(-1到1) 1/[1+t^2] dt
=arctant | (-1到1)
=π/2