积分：求√θ×(cos(θ^3/2))^2dθ在0到π^2/3的积分.要用替换法/换元法算.

Question

积分：求√θ × (cos(θ^3/2))^2 dθ在0到π^2/3的积分.要用替换法/换元法算.

Answer 1

令 u=θ^(3/2),du = (3/2)√θ dθ
原式= ∫ (2/3) (cosu)^2 du u:[0,π]
= (1/3) ∫ [0,π] ( 1+ cos 2u ) du
= (1/3) [ u + sin(2u) /2 ] |(u=π) - (1/3)[ u + sin(2u) /2 ] |(u=0)
= π/3