Question

Please solve this with proper steps ​

Answer 1

$\: \bold \green{ || \: || \: your \: answer || \: || }$

$= \int \frac{1}{1 + \sin( \alpha ) } dx \\ \\ = \int \frac{1 - \sin( \alpha ) }{(1 + \sin( \alpha ) ) \times (1 - \sin( \alpha )) } dx \: \: \: \bold \red{(: rationalised)} \\ \\ = \int \frac{1 - \sin( \alpha ) }{1 - \sin {}^{2} ( \alpha ) } dx \\ \\ = \int \frac{1 - \sin( \alpha ) }{ \cos {}^{2} ( \alpha ) } dx \\ \\ = \int (\frac{1}{ \cos {}^{2} ( \alpha ) } - \frac{ \sin( \alpha ) }{ \cos {}^{2} ( \alpha ) } ) \: dx \\ \\ \: = \int( \sec {}^{2} ( \alpha ) - \tan( \alpha ) \sec( \alpha ))dx \\ \\ \bold \green{ = ( \tan( \alpha ) - \sec( \alpha ) + c )} \: \boxed{ans}$

$\: \: \bold \orange{till \: then \: be \: happy.........}$