微积分求教

1.
令e^t - 1 = u²，e^tdt = 2udu
即(1+u²)dt = 2udu
则dt/√(e^t-1) = dt/u = 2du/(1+u²)
当ln2 < t < 2ln2时
1 < u < √3
原式=∫(1,√3)2du/(1+u²) = 2arctan u|(1,√3) = 2arctan √3 - 2arctan 1 = 2π/3 - π/2 = π/6

2.
1+cosx = 2cos²(x/2)
sinx/(1+cosx) = tan(x/2)
x/(1+cosx) = x/2cos²(x/2) = x/2 * sec²(x/2) = x * [tan(x/2)]'
故(x+sinx)/(1+cosx) = tan(x/2) + x * [tan(x/2)]' = [xtan(x/2)]'
原式=[xtan(x/2)]|(0,π/2) = π/2

3.
φ'x = y * x^(y-1) + 2x
φ'x(1,1) = 3
φ''yx = [x^y * lnx - 1]'x = y * x^(y-1) * lnx + x^y/x = x^(y-1) * (ylnx + 1)
φ''yx(1,1) = 1

4.
∂φ/∂x = ∂φ/∂u * ∂u/∂x + ∂φ/∂v * ∂v/∂x = 2ulnv * 1/y + u²/v * 3 = 2xln(3x-2y) / y² + 3x²/[y²(3x-2y)]
∂φ/∂y = ∂φ/∂u * ∂u/∂y + ͦ