Question

Guy solve this also please.....​

Answer 1

Step-by-step explanation:

We have,

$\int \frac{dx}{3 + 2 \cos(x) }$

$= \int \frac{dx}{3 + 2( \frac{1 - \tan^{2} ( \frac{x}{2} ) }{1 + \tan^{2} ( \frac{x}{2} ) }) }$

$= \int \frac{(1 + \tan^{2} ( \frac{x}{2} ) )dx}{5 + \tan^{2} ( \frac{x}{2} ) }$

$= \int \frac{ \sec^{2} ( \frac{x}{2} ) dx}{5 + \tan^{2} ( \frac{x}{2} ) }$

Let tan(x/2) = t

=> 1/2•sec²(x/2) dx = dt

=> sec²(x/2) dx = 2dt

$= 2 \int \frac{dt}{ {t}^{2} + { (\sqrt{5} )}^{2} }$

$= 2. \frac{1}{ \sqrt{5} } \tan^{ - 1} ( \frac{t}{ \sqrt{5} } ) + c$

$= > \frac{2}{ \sqrt{5} } \tan^{ - 1} ( \frac{ \tan( \frac{x}{2} ) }{ \sqrt{5} } ) + c$