I think that this is the first wime when I handle an integration with infinitesimal(differetial?) element with combined function .
$$ a \in \mathbb R_{> 0} $$
$$ \int_{0 }^{\frac{\pi}{2} } \frac{ 1 }{ a ^{2} + \tan^{2}\left( x \right) } \,d \left( \tan^{}\left( x \right) \right) $$
I've done the below transformations .
$$ \int_{0 }^{\frac{\pi}{2} } \frac{ 1 }{ a ^{2} + \tan^{2}\left( x \right) } \,d \left( \tan^{}\left( x \right) \right) $$
$$ = \left( \frac{ \frac{ a }{ a } }{ \frac{ 1 }{ 1 } } \right) \int_{0 }^{\frac{\pi}{2} } \frac{ 1 }{ a ^{2} + \tan^{2}\left( x \right) } \,d \left( \tan^{}\left( x \right) \right) $$
$$ = \left( \frac{ \frac{ 1 }{ a } }{ \frac{ 1 }{ 1 } } \right) \int_{0 }^{\frac{\pi}{2} } \frac{ a }{ a ^{2} + \tan^{2}\left( x \right) } \,d \left( \tan^{}\left( x \right) \right) $$
$$ = \frac{1}{ a } \int_{0 }^{\frac{\pi}{2} } \frac{ a }{ a ^{2} + \tan^{2}\left( x \right) } \,d \left( \tan^{}\left( x \right) \right) $$
$$ = \frac{1}{ a } \left[ \tan^{-1} \left( \frac{ \tan^{}\left( x \right) }{ a } \right) \right]_{0}^{\frac{\pi}{2} } $$
$$ = \frac{ \pi }{ 2a } $$
Currently I can't get why this integration is correct .
First things to first , can I assume that the values of $~ d \left( \tan^{}\left( x_{} \right) \right) ~$ are all same for all $~ x ~$ in the limit of $~ \left[ 0:\frac{\pi}{2} \right] ~$?
HINT:
Note upper limit of $\tan(x)=u $ set conveniently has $ \infty$ for the upper limit but not $\pi/2$.
So
$$ \int_{0 }^{\infty } \frac{ du }{ a ^2 + u^2} = \frac{1}{a}\; \tan^{-1}\frac{u}{a}+c $$
and evaluating the two limits:
$$=\frac{1}{a}\; (\tan^{-1}\infty -0)=\frac{\pi}{2a}.$$