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# Integrate the function$\frac{\cos x}{\sqrt{1+\sin x}}$

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Toolbox:
• Method of substitution :
• Given f(x)dx can be transformed into another form by changing independent variable x to t by substituting x=g(t).
• Consider $I=\int f(x)dx.$
• Put x=g(t) so that $\frac{dx}{dt}=g'(t).$
• $\Rightarrow$dx=g'(t)dt.
• Thus $I=\int f(g(t).g'(t))dt.$
Given $I=\int \frac{\cos x}{\sqrt {1+\sin x}}dx.$

Put $t=1+\sin x.$

$\;\;\;dt=\cos xdx.$

Now substituting for x and dx we get,

$I=2\int\frac{dt}{\sqrt t}=\int t^{\frac{-1}{2}}dt.$

On integrating we get,

$\;\;\;=\begin{bmatrix}\frac{t^\frac{-1}{2}+1}{\frac{-1}{2}+1}\end{bmatrix}=2(\sqrt t).$

Substituting back for t we get,

$\int \frac{\cos x}{\sqrt {1+\sin x}}dx=2\sqrt{1+\sin x}+c.$

answered Jan 29, 2013

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