Question

求解一道定积分题

求定积分 上是3/4拍下是负3/4拍
(1+arctan x)乘以根号1+cosx dx

Answer 1

arctan x*√(1+cosx)是奇函数，故
原式=∫[-3π/4,3π/4]√(1+cosx)dx
=√2∫[-3π/4,3π/4](cos(x/2)dx
=2√2sin(x/2)|[-3π/4,3π/4]
=4√2sin(3π/8)
=2√(4+2√2)