Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Provide Solution For RD Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.23 Question 10 Maths Textbook Solution.

Provide Solution For  RD Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.23 Question 10 Maths Textbook Solution.

Answers (1)

Answer : \frac{2}{3} \tan ^{-1}\left[3 \tan \frac{x}{2}\right]+c

Hint: To solve this equation we have to change cosx into tanx form

Given: \int \frac{1}{5-4 \cos x} d x

Solution :\int \frac{1}{5-4 \cos x}dx

\operatorname{Cos} x=\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}

\begin{aligned} &=\int \frac{1}{5-4\left(\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\right)} \mathrm{d} x \\ &= \\ &=\int \frac{1+\tan ^{2} \frac{x}{2}}{1+9 \tan ^{2} \frac{x}{2}} d x \end{aligned}

\begin{aligned} &=\int \frac{\sec ^{2} \frac{x}{2}}{1+9 \tan ^{2} \frac{x}{2}} d x \\ &=\int \frac{\sec ^{2} \frac{x}{2}}{1+\left(3 \tan \frac{x}{2}\right)^{2}} d x \end{aligned}

\begin{aligned} &\text { Let } 3 \tan \frac{x}{2}=t \\ &=3 \sec ^{2} \frac{x}{2} \cdot \frac{1}{2} d x=d t \end{aligned}

\begin{aligned} &=\frac{2}{3} \int \frac{d t}{1+t^{2}} \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left\langle\int\frac{1}{a^{2}+t^{2}}dt=\frac{1}{a} \tan ^{-1}\left(\frac{t}{a}\right)\right\rangle \\ &=\frac{2}{3} \tan ^{-1}(t)+C \\ &=\frac{2}{3} \tan ^{-1}\left(3 \tan \frac{x}{2}\right)+C \end{aligned}

Posted by

infoexpert23

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 Class 12 history paper 2025 moderately difficult, tested students’ understanding
anam-khan

CBSE Class 12 Computer Science Exam Analysis 2025: Paper 'moderate', but 'tougher' than last three years
anam-khan

CBSE Class 12 Accountancy Exam Analysis 2025: Paper ‘moderate', includes application-based questions

Latest Question