Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Please solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.23 Question 4 maths textbook solution.

Please solve RD Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.23 Question 4 maths textbook solution.

Answers (1)

Answer : \frac{1}{\sqrt{15}} \log \left|\frac{\sqrt{3}+\sqrt{5} \tan \frac{x}{2}}{\sqrt{3}-\sqrt{5} \tan \frac{x}{2}}\right|+c

Hint: To solve this statement we have to convert cos into tan form

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

Solution :

\int \frac{1}{4 \cos x-1} d x

Use the formula : \left[\cos x=\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\right]

\begin{aligned} &=\int \frac{1}{4\left(\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\right)-1} d x \\ &=\int \frac{1+\tan ^{2} \frac{x}{2}}{4-4 \tan ^{2} \frac{x}{2}-1-\tan ^{2} \frac{x}{2}} d x \\ &=\int \frac{\sec ^{2} \frac{x}{2}}{3-5 \tan ^{2} \frac{x}{2}} d x \end{aligned}

\begin{aligned} &{\left[\begin{array}{l} \tan \frac{x}{2}=t \\ \frac{1}{2} \sec ^{2} \frac{x}{2} d x=d t \end{array}\right]} \\ &=\int \frac{2 d t}{3-5 t^{2}} \\ &=\frac{1}{\sqrt{3}} \int \frac{2 \sqrt{3}}{(\sqrt{3}-\sqrt{5} t)(\sqrt{3}+\sqrt{5} t)} d t \\ &=\frac{1}{\sqrt{3}} \int \frac{(\sqrt{3}-\sqrt{5} t)+(\sqrt{3}+\sqrt{5} t)}{(\sqrt{3}-\sqrt{5} t)(\sqrt{3}+\sqrt{5} t)} d t \end{aligned}

\begin{aligned} &=\frac{1}{\sqrt{3}} \int \frac{d t}{\sqrt{3}+\sqrt{5} t}+\frac{1}{\sqrt{3}} \int \frac{d t}{\sqrt{3}-\sqrt{5} t}\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \quad\left[\int \frac{d t}{a t+b}=\log \left|\frac{a t+b}{a}\right|+c\right] \\ &=\frac{1}{\sqrt{3}} \log \left|\frac{\sqrt{3}+\sqrt{5} t}{\sqrt{5}}\right|+\frac{1}{\sqrt{3}} \log \left|\frac{\sqrt{3}-\sqrt{5} t}{-\sqrt{5}}\right|+c \end{aligned}

\begin{aligned} &=\frac{1}{\sqrt{3}} \times \frac{1}{\sqrt{5}} \log \left|\frac{\sqrt{3}+\sqrt{5} t}{\sqrt{3}-\sqrt{5} t}\right|+c \\ &=\frac{1}{\sqrt{15}} \log \left|\frac{\sqrt{3}+\sqrt{5} \tan \frac{x}{2}}{\sqrt{3}-\sqrt{5} \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 Date Sheet 2027 Live: Class 10, 12 exam dates soon; LOC submission process, paper pattern
anam-khan

CBSE Exam Date 2027 Highlights: Class 10, 12 exam dates soon on cbse.gov.in; private candidate form
anam-khan

CBSE 10th, 12th board exam schedule 2027: Sample papers, passing marks

Latest Question