1.∫(- 1→1) √(1 - x)√(1 + x) dx
= ∫(- 1→1) √(1 - x²) dx
= 1/2 * π(1)²
= π/2
2.∫(- 1→1) x³e^(x²) dx
= 0
3.∫(0→1) √(x² + 1) dx,x = tanz,dx = sec²z dz
= ∫(0→π/4) √(tan²z + 1) * sec²z dz
= ∫(0→π/4) sec³z dz
= (1/2)secztanz + (1/2)ln(secz + tanz) |(0→π/4)
= (1/2)(√2)(1) + (1/2)ln(√2 + 1)
= 1/√2 + (1/2)ln(1 + √2)
4.∫(0→π) 1/(2 - cosx) dx
= ∫(0→π) 1/[2sin²(x/2) + 2cos²(x/2) - cos²(x/2) + sin²(x/2)] dx
= ∫(0→π) 1/[3sin²(x/2) + cos²(x/2)] dx
= 2∫(0→π) 1/[1 + 3tan²(x/2)] d[tan(x/2)]
= (2/√3)arctan[√3tan(x/2)] |(0→π)
= (2/√3)(π/2)
= π/√3
5.∫(-∞→+∞) e^(- x²/2) dx
= ∫(-∞→+∞) e^(- (x/√2)²) dx
u = x/√2,dx = √2 du
= √2∫(-∞→+∞) e^(- u²) du
= √2 * √π
= √(2π)
你自己说了不要解释的,别又问长问短了.