Question

计算下列定积分

Answer 1

∫(1,4)(√x-1)/xdx
=∫(1,4)1/√xdx-∫(1,4)1/xdx
=2√x|(1,4)-lnx|(1,4)
=2(√4-√1)-(ln4-ln1)
=2(2-1)-(2ln2-0)
=2-2ln2
∫(2,4)|x-3|dx
=∫(2,3)(3-x)dx+∫(3,4)(x-3)dx
=(3x-1/2x^2)|(2,3)+(1/2x^2-3x)|(3,4)
=3(3-2)-1/2(3^2-2^2)+1/2(4^2-3^2)-3(4-3)
=3-1/2(9-4)+1/2(16-9)-3
=-5/2+7/2
=1
∫(-1,1)x/√(5-4x)dx
令u=√(5-4x)
u1=√[5-4×(-1)]=3
u2=√(5-4×1)=1
x=(5-u^2)/4
dx=-u/2du
原式=∫(3,1)(5-u^2)/(4u)×(-u/2)du
=1/8∫(3,1)(u^2-5)du
=1/8(1/3u^3-5u)|(3,1)
=1/8[1/3(1^3-3^3)-5(1-3)]
=1/8(-26/3+10)
=1/8×4/3
=1/6
∫(1,e)(1+lnx)dx
=∫(1,e)dx+∫(1,e)lnxdx
=x|(1,e)+xlnx|(1,e)-∫(1,e)xdlnx
=(e-1)+(elne-1ln1)-∫(1,e)dx
=e-1+e-0-(e-1)
=e