Question

integration of 1/1-Sinx .dx

integration of (√x+1/√x)² dx ​

Answer 1

$\large \bigstar \underline{ \pmb{ \orange{{ \frak{solution \: 1 : - }}}}}$

$\\ : \implies \displaystyle \int \bigg(\sf\dfrac{1}{1 - \sin x} \bigg).dx$

$\\ : \implies \displaystyle \int \bigg[\bigg( \sf \frac{1}{1 - \sin x} \bigg) \times \bigg( \frac{1 + \sin x}{1 + \sin x} \bigg) \bigg] dx$

$\\ : \implies \displaystyle \int \sf \dfrac{1 + \sin x}{1 - { \sin}^{2}x } dx$

$\\ : \implies \displaystyle \int \sf \frac{1 + \sin x}{ { \cos}^{2}x } dx$

$\\ : \implies \displaystyle \int \sf { \sec}^{2} x + \tan x. \sec x.dx$

$\\ : \implies \boxed{ \sf \tan x + \sec x + c} \bigstar$

$\large \bigstar \underline{ \pmb{ \orange{{ \frak{solution \: 2: - }}}}}$

$\\ : \implies \displaystyle \int \sf \bigg( \sqrt{x} + \frac{1}{ \sqrt{x} } \bigg)^{2} dx$

$\\ : \implies \displaystyle \int \sf \bigg(x + \frac{1}{x} + 2 \bigg)dx$

$\\ : \implies \boxed{ \sf \frac{ {x}^{2} }{2} + lnx + 2x + c} \bigstar$