Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Explain Solution R D Sharma Class 12 Chapter 19 Definite Integrals Exercise 19 point 1 Question 36 Maths Textbook Solution

Explain Solution R.D.Sharma Class 12 Chapter 19 Definite Integrals  Exercise 19.1 Question 36 Maths Textbook Solution.

Answers (1)

Answer:

            \frac{e^{2}}{2}-e

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

Given:

            \int_{e}^{e^{2}}\left \{ \frac{1}{log\: x}-\frac{1}{\left ( log\: x \right )^{2}} \right \}dx

Solution:

\begin{aligned} &\int_{e}^{e^{2}}\left\{\frac{1}{\log x}-\frac{1}{(\log x)^{2}}\right\} d x=\int_{e}^{e^{2}} \frac{1}{\log x} d x-\int_{e}^{e^{2}} \frac{1}{(\log x)^{2}} d x \\ &=\int_{e}^{e^{2}} \frac{1}{\log x} \cdot 1 d x-\int_{e}^{e^{2}} \frac{1}{(\log x)^{2}} d x \\ &=\left[\frac{1}{\log x} \int 1 d x\right]_{e}^{e^{2}}-\int_{e}^{e^{2}}\left\{\frac{d}{d x}\left(\frac{1}{\log x}\right) \int 1 d x\right\} d x-\int_{e}^{e^{2}} \frac{1}{(\log x)^{2}} d x \\ &\quad\left[\because \int 1 d x=x \& \frac{d}{d x}\left(\frac{1}{x}\right)=\frac{d}{d x}\left(x^{-1}\right)=-1 x^{-1-1}=-1 \cdot x^{-2}=-\frac{1}{x^{2}}\right] \end{aligned}

\begin{aligned} &=\left[\frac{1}{\log x} x\right]_{\theta}^{e^{2}}-\int_{\theta}^{e}\left(-\frac{1}{(\log x)^{2}}\right) \frac{1}{x} x d x-\int_{e}^{e^{2}} \frac{1}{(\log x)^{2}} d x \\ &=\left[\frac{x}{\log x}\right]_{e}^{e^{2}}+\int_{e}^{e^{2}} \frac{1}{(\log x)^{2}} d x-\int_{e}^{e^{2}} \frac{1}{(\log x)^{2}} d x \\ &=\left[\frac{e^{2}}{\log e^{2}}-\frac{e}{\log e}\right] \\ &=\left[\frac{e^{2}}{2 \log e}-\frac{e}{\log e}\right] \; \; \; \; \; \; \; \; \; \; \; \quad[\because \log e=1] \\ &=\left[\frac{e^{2}}{2}-e\right] \end{aligned}

Posted by

infoexpert21

View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support
cuet_ads

Similar Questions

Trending Articles/News

anam-khan

CBSE Board Exam Dates 2027 LIVE: CBSE 10th, 12th time table soon; guidelines, changes announced this year
anam-khan

CBSE invites LOC submission for Class 12 re-exams 2026 in West Asian countries
anam-khan

CBSE Datesheet 2027 LIVE: Class 10, 12 board exam timetable soon; sample question papers, marking scheme

Latest Question