Question

计算积分:

Answer 1

π*∫dx/{1+[cos(x)]^2}

=π*∫[sec(x)]^2dx/{1+[sec(x)]^2}

=π*∫[sec(x)]^2dx/{2+ [tan(x)]^2}

=π*∫2^(-1/2)d[tan(x)/2^(1/2)]/{1+[tan(x)/2^(1/2)]^2}

=π*2^(-1/2)arctan[tan(x)/2^(1/2)]+C

=π*[(√2)/2]*arctan{[(√2)/2]*tan(x)}+C

C 为任意常数。

I=π*[(√2)/2]*arctan{[(√2)/2]*tan(π/2)}

-π*[(√2)/2]*arctan{[(√2)/2]*tan(0)}

=((√2)*π^2)/4 

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另附MATLAB运行结果，可以知道笔算结果是对的。

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