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  4. Evaluate: ∫ limits0π/2(1/2 cos x +4 sin x) dx

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Q. Evaluate: $ \int\limits_{0}^{{\pi}/{2}}\frac{1}{2\, cos\,x +4\,sin\,x} dx$

Integrals

Solution:

We have, $I= \int\limits_{0}^{\pi/2}\frac{1}{2 \,cos \,x +4\,sin \,x} dx$
$\Rightarrow I = \int\limits_{0}^{\pi/2}\frac{1}{2 \left(\frac{1-tan^{2} \frac{x}{2}}{1+tan^{2} \frac{x}{2}}\right) + 4\left(\frac{2 tan \frac{x}{2}}{1+ tan^{2} \frac{x}{2}}\right)} dx $

$ \Rightarrow I = \int\limits_{0}^{\pi /2}\frac{sec^{2} \frac{x}{2}}{2- 2 tan^{2 } \frac{x}{2} + 8 tan \frac{x}{2}} dx$

Let tan $\frac{x}{2} =t \Rightarrow \frac{1}{2} sec^{2} \frac{x}{2} dx = dt $

Also, $x= 0 \Rightarrow t = 0 and x = \frac{\pi}{2} \Rightarrow t = 1$

$ \therefore I = \int\limits_{0}^{1} \frac{2dt}{2 - 2t^{2} +8t} $

$ = \int\limits_{0}^{1} \frac{dt}{-\left[\left(t-2\right)^{2}-5\right]} = \int\limits_{0}^{1} \frac{dt}{\left(\sqrt{5}\right)^{2}-\left(t-2\right)^{2}} dt $

$ \Rightarrow I = \frac{1}{2\sqrt{5}} \left[log\left|\sqrt{5}+t -2\right|\right]_{0}^{1} $

$ = \frac{1}{2\sqrt{5}} \left[log\left(\frac{\sqrt{5}-1}{\sqrt{5}+1}\right) -log\left(\frac{\sqrt{5}-2}{\sqrt{5}+2}\right)\right]$

$= \frac{1}{2\sqrt{5}} log\left(\frac{3+\sqrt{5}}{3-\sqrt{5}}\right) = \frac{1}{2\sqrt{5}} log\left(\frac{3+\sqrt{5}}{2}\right) ^{2}$

$ \Rightarrow I = \frac{1}{\sqrt{5}} log\left(\frac{3+\sqrt{5}}{2}\right)$