积分上限1,积分下限-1,x/根号下5-4Xdx的定积分的解答过程

题目

积分上限1,积分下限-1,x/根号下5-4Xdx的定积分的解答过程

答案

∫(-1,1) x/√(5-4x) dx
Let z=5-4x => x=(5-z)/4 and dx=(-1/4) dz
When x=-1,z=9 // when x=1,z=1
So the integral = ∫(9,1) (5-z)/4 * 1/√z * (-1/4) dz
= (1/16)∫(1,9) (5-z)/√z dz
= (5/16)∫(1,9) 1/√z dz - (1/16)∫(1,9) √z dz
= (5/16)*2√z[1,9] - (1/16)*(2/3)*z^(3/2)[1,9]
= (5/8)(3-1) - (1/24)(27-1)
= 5/4 - 13/12
= 1/6