1.

Dx/√(1+sinx). find the integral

Answer»

Given, \bold{\int{\frac{dx}{\sqrt{<klux>1</klux>+<klux>SINX</klux>}}}}

First of all we have to resolve \bold{\sqrt{1+sinx}}
sinx = COS(π/2 - x)
∴ 1 + sinx = 1 + cos(π/2 - x)
But we know, 1 + cosФ = 2cos²Ф/2 , use it here,
∴ 1 + cos(π/2 - x) = 2cos²(π/2 - x)/2
= 2cos²(π/4 - x/2)

HENCE, \bold{\sqrt{1+sinx}=\sqrt{2}cos(\frac{\pi}{4}-\frac{x}{2})}
So, integration CONVERTS in \bold{\int{\frac{dx}{\sqrt{2}cos(\pi/4-x/2)}}}
= \bold{\frac{1}{\sqrt{2}}\int{sec(\pi/4-x/2).dx}}
= \bold{\frac{-2\times1}{\sqrt{2}}log|tan(\pi/4-x/2)+sec(\pi/4-x/2)|} +C
= \bold{-\sqrt{2}log|tan(\pi/4-x/2)+sec(\pi/4-x/2)|} +C


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