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  • Please solve RD Sharma class 12 Chapter Definite integrals exercise 19.2 question 8 maths textbook solution.

Please solve RD Sharma class 12 Chapter Definite integrals exercise 19.2 question 8 maths textbook solution.

Answers (1)

\sin (\log 3)

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

Given: \int_{1}^{3}\frac{\cos (\log x)}{x}dx

Solution: \int_{1}^{3}\frac{\cos (\log x)}{x}dx

Put \log x=t

\frac{1}{x} dx=dt

dx=x dt

When x=1  then t=log1=0  and when x=3  then t=log3

\begin{aligned} &=\int_{0}^{\log 3} \frac{\cos t}{x} x d t \\ &=\int_{1}^{\log 3} \cos t d t\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\int \cos x d x=\sin x+c\right] \end{aligned}

\begin{aligned} &=[\sin t]_{0}^{\log 3} \\ &=[\sin (\log 3)-\sin 0] \\ &=\sin (\log 3) \end{aligned}

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