Question

定积分中的三角函数变换问题？

请问高手，图中的x/(1+cosx)如何转换为x/2*sec^2(x/2)，就是图中红色下划线部分

Answer 1

倍角公式 1+cos2u = 2(cosu)^2
则 ∫xdx/(1+cosx) = ∫xdx/[2(cosx/2))^2]
=∫(x/2)(secx/2)^2dx = ∫x(secx/2)^2d(x/2)
= ∫xdtan(x/2) = xtan(x/2) - ∫tan(x/2)dx
= xtan(x/2) - ∫tan(x/2)dx
= xtan(x/2) - 2∫[sin(x/2)/cos(x/2)]d(x/2)
= xtan(x/2) - 2∫[-1/cos(x/2)]dcos(x/2)
= xtan(x/2) + 2lncos(x/2), 代入数值即可。

Answer 2

二倍角公式,1+cosx = 2(cos(x/2))^2 = 2/(sec(x/2))^2