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 Class 12 Chapter 19 Definite Integrals Exercise Revision Exercise Question 58

Provide Solution for RD Sharma Class 12 Chapter 19 Definite Integrals Exercise Revision Exercise Question 58

Answers (1)

best_answer

Answer:  \frac{-1}{2 \sqrt{5}} \log \left[\frac{2 \sqrt{5}-\sqrt{5}-3}{2 \sqrt{5}+\sqrt{5}-3}\right]

Hint: To solve this we convert  sin and cos into tan

Given:

\int_{0}^{\frac{\pi}{2}} \frac{1}{2 \cos x+4 \sin x} d x

Solution:

Let

\begin{aligned} &I=\int_{0}^{\frac{\pi}{2}} \frac{1}{2 \cos x+4 \sin x} d x \\ & \end{aligned}

I=\frac{1}{2} \int_{0}^{\frac{\pi}{2}} \frac{1}{\cos x+2 \sin x} d x

 

=\frac{1}{2} \int_{0}^{\frac{\pi}{2}} \frac{\sec ^{2} \frac{x}{2}}{1-\tan ^{2} \frac{x}{2}+4 \tan ^{2} \frac{x}{2}} d x

Let

\begin{aligned} &t=\tan ^{2} \frac{x}{2} \\ & \end{aligned}

\frac{d t}{d x}=\sec ^{2} \frac{x}{2} \times \frac{1}{2}

2 d t=\sec ^{2} \frac{x}{2} d x

Now

\begin{aligned} &\frac{1}{2} \times 2 \int_{0}^{1} \frac{d t}{1-t^{2}+4 t} \\ & \end{aligned}

I=-\int_{0}^{1} \frac{d t}{t^{2}-4 t-1} \\

I=-\int_{0}^{1} \frac{d t}{t^{2}-4 t-1+2^{2}-2^{2}}

\begin{aligned} &I=-\int_{0}^{1} \frac{d t}{(t-2)^{2}-(\sqrt{5})^{2}} \\ & \end{aligned}

I=\frac{-1}{2 \sqrt{5}} \log \left|\frac{t-2-\sqrt{5}}{t-2+\sqrt{5}}\right|_{0}^{1} \\

I=\frac{-1}{2 \sqrt{5}} \log \left|\frac{-1-\sqrt{5}}{-1+\sqrt{5}}\right|-\log \left|\frac{-2-\sqrt{5}}{-2+\sqrt{5}}\right|

\begin{aligned} &=\frac{-1}{2 \sqrt{5}} \log \left[\left|\frac{-1-\sqrt{5}}{-1+\sqrt{5}}\right| \times\left|\frac{-2-\sqrt{5}}{-2-\sqrt{5}}\right|\right] \\ & \end{aligned}

=\frac{-1}{2 \sqrt{5}} \log \left(\frac{2-\sqrt{5}+2 \sqrt{5}-5}{2+\sqrt{5}-2 \sqrt{5}-\sqrt{5}}\right) \\

=\frac{-1}{2 \sqrt{5}} \log \left(\frac{2 \sqrt{5}-\sqrt{5}-3}{-2 \sqrt{5}+\sqrt{5}-3}\right)

 

Posted by

infoexpert27

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 2026: Exam form submission for Class 10, 12 private students begins tomorrow
anam-khan

CBSE launches integrated payment system for Class 12 practical exam remuneration
anam-khan

CBSE Board Exams 2026: Online portal for availing CWSN facilities opens from September 9

Latest Question