Question

This is engineering mathematics 2 question from multivariable calculus 2​

Answer 1

Step-by-step explanation:

We have,

$\int \limits^{ \infty }_{0} \frac{dx}{(1 + {x}^{2})(1 + \tan ^{ - 1} (x)) }$

$Let \: 1 + \tan^{ - 1} (x) = t \\ \implies \frac{dx}{1 + {x}^{2} } = dt$

$= \int \limits^{ 1 + \frac{\pi}{2} }_{ 1 } \frac{dt}{t }$

$= [ ln(t) ]^{1 + \frac{\pi}{2} } _{1}$

$= ln(1 + \frac{\pi}{2} ) - ln(1)$

$= ln(1 + \frac{\pi}{2} )$