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  • Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 14.

Provide solution for RD Sharma maths class12 Chapter Definite Integrals exercise 19.2 question 14.

Answers (1)

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

Hint: We use indefinite integral formula then put limits to solve this integral.

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

Solution: I=\int_{0}^{\frac{\pi}{3}}\frac{\cos x }{3+4 \sin x}dx

Put 3+4\sin x=t

4\cos x \; dx=dt

\cos x\; \; dx=\frac{dt}{4}

When x=0  then t=3  and

when x=\frac{\pi}{3}  then t=3+2\sqrt{3}

\begin{aligned} &I=\int_{3}^{3+2 \sqrt{3}} \frac{1}{t} \frac{d t}{4} \\ &I=\frac{1}{4} \int_{3}^{3+2 \sqrt{3}} \frac{1}{t} d t \\ &=\frac{1}{4}[\log |t|]_{3}^{3+2 \sqrt{3}} \\ &=\frac{1}{4}[\log (3+2 \sqrt{3})-\log 3] \\ &=\frac{1}{4} \log \left(\frac{3+2 \sqrt{3}}{3}\right) \end{aligned}

 

 

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