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  • Explain Solution R.D.Sharma Class 12 Chapter 19 Definite Integrals Exercise 19.1 Question 35 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 19 Definite Integrals  Exercise 19.1 Question 35 Maths Textbook Solution.

Answers (1)

Answer:

            \frac{1}{2}

Hint: Use indefinite formula and put limits to solve this integral

Given: 

\int_{1}^{e}\frac{log\: x}{x}dx

Putting log x=t

            \Rightarrow \frac{1}{x}dx=dt

\Rightarrow dx=xdt

and when x=1 thent=log1=0

                                                       \left [ \because log\; 1=0 \right ]

When x=e then t=loge=1

                                                        \left [ \because log\: e=1 \right ]

Then

\begin{aligned} &\int_{1}^{e} \frac{\log x}{x} d x=\int_{0}^{1} \frac{t}{x} x d t \\ &=\int_{0}^{1} t d t \\ &=\left[\frac{t^{1+1}}{1+1}\right]_{0}^{1} \\ &=\left[\frac{t^{2}}{2}\right]_{0}^{1} \end{aligned}

\begin{aligned} &=\frac{1}{2}\left[t^{2}\right]_{0}^{1} \\ &=\frac{1}{2}\left[1^{2}-0^{2}\right] \\ &=\frac{1}{2}[1-0] \\ &=\frac{1}{2} \end{aligned}

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