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定积分1到4+分之1+根号下5-4x分之x的定积分
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亲,你好!非常高兴为你解答。
解析:
设 根号下(5 - 4x) = t
则 (5 - 4x) = t^2
x = (5 - t^2) / 4
dx = - (1/2)tdt
∫x/根号下(5 - 4x) dx
= ∫[(5 - t^2) / 4] (1/t) [- (1/2)tdt]
= -(1/8) ∫ (5 - t^2) dt
= -(1/8) [5t - (1/3) t^3] + c
= -(1/8) {5根号下(5 - 4x) - (1/3) [(根号下(5 - 4x))的立方]} + c
= -(1/8) {5(5 - 4x)^(1/2) - (1/3) (5 - 4x)^(3/2)} + c
咨询记录 · 回答于2023-12-22
定积分1到4+分之1+根号下5-4x分之x的定积分
亲,你好!非常高兴为你解答。
解析:设 根号下(5 - 4x) = t
则 (5 - 4x) = t^2
x = (5 - t^2) / 4
dx = - (1/2)tdt
∫ x/根号下(5 - 4x) dx
= ∫ [(5 - t^2) / 4] (1/t) [- (1/2)tdt]
= - (1/8) ∫ (5 - t^2) dt
= - (1/8) [5t - (1/3)t^3] + c
= - (1/8) {5根号下(5 - 4x) - (1/3) [(根号下(5 - 4x))的立方]} + c
= - (1/8) {5(5 - 4x)^(1/2) - (1/3) (5 - 4x)^(3/2)} + c
dy/dx=2xyy'=2xyy'/y=2x(lny)'=2x积分:lny=x^2+lnCln(y/C)=x^2y=Ce^(x^2)x=0时:y=C=1所以:特解为y=e^(x^2)