dx\(1+cos^2x)从0到派\2的定积分
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/27 14:19:56
![dx\(1+cos^2x)从0到派\2的定积分](/uploads/image/z/3892253-5-3.jpg?t=dx%5C%281%2Bcos%5E2x%29%E4%BB%8E0%E5%88%B0%E6%B4%BE%5C2%E7%9A%84%E5%AE%9A%E7%A7%AF%E5%88%86)
dx\(1+cos^2x)从0到派\2的定积分
dx\(1+cos^2x)从0到派\2的定积分
dx\(1+cos^2x)从0到派\2的定积分
∫(0→π/2) dx/(1 + cos^2x)
= ∫(0→π/2) dx/[(sin^2x + cos^2x) + cos^2x]
= ∫(0→π/2) dx/(sin^2x + 2cos^2x)
= ∫(0→π/2) dx/[cos^2x(2 + tan^2x)]
= ∫(0→π/2) d(tanx)/(2 + tan^2x)
= (1/√2)arctan[(tanx)/√2] |(0→π/2)
= (1/√2)(π/2 - 0)
= π/(2√2)
原函数1/2*tan(x),定积分不收敛。