Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa

Integrate \int \frac{dx}{\sin^{2} x+2\cos^{2} x+2\sin x\cos x}

  • Option 1)

    x + c

  • Option 2)

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

  • Option 3)

    tan ^{-1}\left ( 1+\tan x \right )+C

  • Option 4)

    none of these

 

Answers (1)

best_answer

As we learned,

 

Type of Integration by perfect square -

Integration are in the format 

(i) \int \frac{dx}{a^{2}\sin ^{2}x+2b\sin x\cos x+c^{2}\cos^{2}x}

(ii) \int \frac{dx}{a\cos^{2}x+b} (iii) \int \frac{dx}{a+b\sin^{2}x}

- wherein

Working rule :

Divide by cos^{2}x in each case and pull t=\tan x ,  dt=\sec^{2}xdx

 

  

Divide by cos^{2}x

\int \frac{\sec ^{2}xdx}{\tan ^{2}x+2+2\tan x}

I=\int \frac{\sec ^{2}xdx}{\left (\tan x+1 \right )^{2}+1}\: \Rightarrow  Put tanx=t

                                                            \sec ^{2}xdx=dt

I=\int \frac{dt}{\left ( t+1 \right )^{2}+1}=\tan ^{-1}\left ( \tan x+1 \right )+C  


Option 1)

x + c

Option 2)

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

Option 3)

tan ^{-1}\left ( 1+\tan x \right )+C

Option 4)

none of these

Posted by

prateek

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main 2025 paper 2 final answer key out; NTA drops 4 questions, 2 from BPlanning
anam-khan

JEE Main 2025 session 2 answer key out for BArch, BPlanning; response sheet on jeemain.nta.nic.in
anam-khan

JEE Mains 2025 Paper 2 Answer Key LIVE: BArch final answer key PDF expected soon at jeemain.nta.nic.in

Latest Question