Question

Evaluate the following:​

Answer 1

Step-by-step explanation:

We have,

$\int \frac{dx}{7 + 2 {x}^{2} }$

$let \: \: x = \sqrt{\frac{7}{2} } \tan(y) \\ \implies \: dx = \sqrt{ \frac{7}{2} } \sec^{2} (y) dy$

$\int \frac{ \sqrt{ \frac{7}{2} } \sec^{2} (y) dy}{7 + 2 . \frac{7}{2} { \tan}^{2} (y)}$

$= \frac{1}{ \sqrt{14} } \int \frac{ \sec^{2} (y)dy }{ \sec^{2} (y) }$

$= \frac{1}{ \sqrt{14} } .y + c$

$= \frac{1}{ \sqrt{14} } . \tan^{-1}( \sqrt{ \frac{2}{7} } x ) + c$