Question

高等数学定积分？

请问这三题怎么做？

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Answer 1

y = 2^x/x^2 + arcsin2x = 2^x · x^(-2) + arcsin2x,
y' = 2^xln2 · x^(-2) + 2^x · (-2)x^(-3)+ 2/√(1-4x^2)
= 2^x(xln2-2)/x^3+ 2/√(1-4x^2)

5. y^2 = 2(x-1), 交 x 轴于顶点 (1, 0). x = 3 时, y = ±2.
两边对 x 求导， 得2yy'= 2, y' = 1/y = ±1/2.
因切线关于 x 轴对称， 故计算旋转体体积只考虑切线斜率 k = y' = 1/2 即可。
切线方程 y = (1/2)(x-3)+2 = (1/2)(x+1), 交 x 轴于 (-1, 0). x = 1 时, y = 1.
Vx = (π/3) · 1^2 · 2 + π∫<1, 3>[(1/4)(x+1)^2-2(x-1)]dx
= 2π/3 + π[(1/12)(x+1)^3-(x-1)^2]<1, 3>
= 2π/3 + π(16/3-4-2/3) = 4π/3

∫<-∞, 0>dx/(1-2x)^(3/2) = (-1/2)∫<-∞, 0>d(1-2x)/(1-2x)^(3/2)
= [1/(1-2x)^(1/2)]<-∞, 0> = 1