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I want to show that the following integral is convergent, and evaluate it:

$$\int_0^{\pi/2} \dfrac{1}{7 + \tan(x)} dx$$.

I plugged the limit into mathematica and got "Directed Infinity". Tried the trick of multiplying the integral by $1$ and see if something more elucidating would come up, but nothing.

The book I got this from says that this limit is for "high $T$", which I interpreted as $T$ goes to infinity. Maybe that could be the problem, nonetheless, I don't see how this limit is done.

Thank you ! :)

egreg
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Tibz
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1 Answers1

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Using Felix Marin's suggestion, you could show that $$\int \dfrac{1}{a + \tan(x)} dx=\frac{a x+\log (a \cos (x)+\sin (x))}{a^2+1}$$ from which it follows that $$\int_0^{\pi/2} \dfrac{1}{a + \tan(x)} dx=\frac{\pi a-2 \log (a)}{2 a^2+2}$$ which does not make any problem as long as $a$ is greater than $0$.

Claude Leibovici
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