Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions Question 16 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Multiple Choice Questions Question 16 Maths Textbook Solution.

Answers (1)

Answer:

\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{1}{\sqrt{6}} \tan \frac{x}{2}\right)+C

Given:

\int\left(\frac{1}{7+5 \cos x}\right) d x

Hint:

You must know about the derivation of tan x and using \int \frac{1}{1+x^{2}} d x

Explanation:

Let \mathrm{I}=\int \frac{1}{7+5 \cos x} d x

\text { Put } \cos x=\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}

\begin{aligned} \therefore I &=\int \frac{1}{7+5\left(\frac{1-\tan ^{2} \frac{x}{2}}{1+\tan ^{2} \frac{x}{2}}\right)} d x \\ &=\int \frac{1+\tan ^{2} \frac{x}{2}}{7\left(1+\tan ^{2} \frac{x}{2}\right)+5-5 \tan ^{2} \frac{x}{2}} d x \\ &=\int \frac{\sec ^{2} \frac{x}{2}}{12+2 \tan ^{2} \frac{x}{2}} \\ &=\frac{1}{2} \int \frac{\sec ^{2} \frac{x}{2}}{\tan ^{2} \frac{x}{2}+6} d x \\ &=\frac{1}{2} \int \frac{2 d t}{t^{2}+(\sqrt{6})^{2}} \end{aligned}    \text { [ Put } \left.\tan \frac{x}{2}=t \Rightarrow \sec ^{2} \frac{x}{2} \cdot \frac{1}{2} d x=d t \Rightarrow \sec ^{2} \frac{x}{2} d x=2 d t\right]

        \begin{aligned} &=\int \frac{d t}{t^{2}+(\sqrt{6})^{2}} \\ &=\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{t}{\sqrt{6}}\right)+C \ldots \ldots \ldots \frac{1}{x^{2}+a^{2}}=\frac{1}{a} \tan ^{-1} \frac{x}{a}+c \\ &=\frac{1}{\sqrt{6}} \tan ^{-1}\left(\frac{\tan \frac{x}{2}}{\sqrt{6}}\right)+C \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 practical exams 2026 will be cancelled over non-compliance of guidelines; board issues SOPs

Latest Question