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  • Provide Solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise Revision Exercise Question 60

Provide Solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise Revision Exercise Question 60

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Answer:   \frac{1}{2 \sqrt{5}} \log \left|\frac{1-\frac{1}{2}-\frac{\sqrt{5}}{2}}{1-\frac{1}{2}+\frac{\sqrt{5}}{2}}\right|-\log \left|\frac{1+\sqrt{5}}{1-\sqrt{5}}\right|

Hint: To this equation we convert cos and sin in terms of tan

Given:  \int_{0}^{\frac{\pi}{2}} \frac{d x}{4 \cos x+2 \sin x}

Solution:  I= \int_{0}^{\frac{\pi}{2}} \frac{d x}{4 \cos x+2 \sin x}

\left[\begin{array}{l} \sin x=\frac{2 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}} \\ \cos x=\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}} \end{array}\right]

\begin{aligned} I &=\int_{0}^{\frac{\pi}{2}} \frac{d x}{\left(\frac{4-4 \tan ^{2} \frac{x}{2}}{4+4 \tan ^{2} \frac{x}{2}}\right)+\frac{4 \tan \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}} \\ \end{aligned}

I =\int_{0}^{\frac{\pi}{2}} \frac{\sec ^{2} \frac{x}{2} d x}{4-4 \tan ^{2} \frac{x}{2}+4 \tan \frac{x}{2}}

\begin{aligned} &{\left[\begin{array}{l} \tan \frac{x}{2}=t \\ \frac{1}{2} \sec ^{2} \frac{x}{2} d x=d t \\ \sec ^{2} \frac{x}{2} d x=2 d t \end{array}\right]} \\ & \end{aligned}

I=\int \frac{2 d t}{4-4 t^{2}-4 t} \\

I=-\frac{1}{2} \int \frac{d t}{t^{2}-2 t-1}

\begin{aligned} I &=-\frac{1}{2} \int \frac{d t}{t^{2}+2 t-1} \\ \end{aligned}

I =-\frac{1}{2} \int \frac{d t}{\left(t-\frac{1}{2}\right)^{2}-\left(\frac{\sqrt{5}}{2}\right)^{2}}

\begin{aligned} &I=-\frac{1}{2} \frac{1}{2 \times \frac{\sqrt{5}}{2}} \log \left|\frac{t-\frac{1}{2}-\frac{\sqrt{5}}{2}}{t-\frac{1}{2}+\frac{\sqrt{5}}{2}}\right| \\ & \end{aligned}

I=-\frac{1}{2 \sqrt{5}} \log \left|\frac{\tan \frac{x}{2}-\frac{1+\sqrt{5}}{2}}{\tan \frac{x}{2}-\frac{1-\sqrt{5}}{2}}\right|_{0}^{\frac{\pi}{2}}

\begin{aligned} &\frac{1}{2 \sqrt{5}} \log \left|\frac{1-\frac{1}{2}-\frac{\sqrt{5}}{2}}{1-\frac{1}{2}+\frac{\sqrt{5}}{2}}\right|-\log \left|\frac{1+\sqrt{5}}{1-\sqrt{5}}\right| \\ & \end{aligned}

I=\frac{1}{2 \sqrt{5}} \log \left|\frac{1-\frac{1}{2}-\frac{\sqrt{5}}{2}}{1-\frac{1}{2}+\frac{\sqrt{5}}{2}}\right|-\log \left|\frac{1+\sqrt{5}}{1-\sqrt{5}}\right|

 

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