Question

How do you evaluate the definite integral `int 4secthetatantheta` from `[-pi/3, pi/3]`?

Answer 1

`int _(-pi/3)^(pi/3)4secthetatanthetad(theta) = 0`

Explanation:

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As we know that the derivative of `" "color(red)(d(sectheta)=secthetatantheta(d theta))" "`

Therefore,
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`int_(-pi/3)^(pi/3)4secthetatantheta d(theta)`
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`=4int_(-pi/3)^(pi/3)color(red)((dsectheta)`
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`=4sectheta|_(-pi/3)^(pi/3)`
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`=4sec(pi/3)-4sec(-pi/3)`
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`=4sec(pi/3)-4sec(pi/3)" "` because `" "color(blue)(sec(-alpha)=secalpha)`
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`=0`
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Hence,
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`int _(-pi/3)^(pi/3)4secthetatanthetad(theta) = 0`