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Need solution for RD Sharma Maths Class 12 Chapter 19 Definite Integrals Excercise Revision Exercise Question 10

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Answer:  \frac{1}{4} \log \left(\frac{2 \sqrt{3}+3}{3}\right)

Given:   \int_{0}^{\frac{\pi}{3}} \frac{\cos x}{3+4 \sin x} d x

Hint: Apply Substitution method


\int_{0}^{\frac{\pi}{3}} \frac{\cos x}{3+4 \sin x} d x

Let  3+4 \sin x=t

4 \cos d x=d t   (diff w.r.t x)

\begin{aligned} &\frac{1}{4} \int_{0}^{\frac{\pi}{3}} \frac{1}{t} d t=\frac{1}{4}(\log t)_{0}^{\frac{\pi}{3}} \\ & \end{aligned}

=\frac{1}{4}[\log |3+4 \sin x|]_{0}^{\frac{\pi}{3}}

\begin{aligned} &=\frac{1}{4}\left(\log \left(3+4 \times \frac{\sqrt{3}}{2}\right)-\log 3\right) \\ & \end{aligned}

=\frac{1}{4}(\log (3+2 \sqrt{3})-\log 3) \\

=\frac{1}{4} \log \left(\frac{3+2 \sqrt{3}}{3}\right)

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