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

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

Solution:

\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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