Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 11

explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 11

Answers (1)

best_answer

Answer:

            \log \left|\frac{(\sin x+2)^{4}}{(\sin x+1)^{2}}\right|+C

Hint:

            To solve this integration, we use partial fraction method   

Given:

            \int \frac{\sin 2 x}{(1+\sin x)(2+\sin x)} d x \\

Explanation:

Let

\begin{aligned} &I=\int \frac{\sin 2 x}{(1+\sin x)(2+\sin x)} d x \\ &I=\int \frac{2 \sin x \cos x}{(1+\sin x)(2+\sin x)} d x \quad[\sin 2 A=2 \sin A \cos A] \end{aligned}             

Let

\begin{aligned} &\sin x=y \\ &\cos x d x=d y \\ &I=\int \frac{2 y d y}{(1+y)(2+y)} \end{aligned}

\begin{aligned} &\frac{2 y}{(1+y)(2+y)}=\frac{A}{1+y}+\frac{B}{2+y} \\ &\frac{2 y}{(1+y)(2+y)}=\frac{A(2+y)+B(1+y)}{(1+y)(2+y)} \\ &2 y=(2 A+B)+y(A+B) \end{aligned}

Comparing coefficient

2=A+B                    (1)

2A+B=0                  (2)

Subtract equation (1) from equation (2)

A=-2

Equation (1)

2=-2+B\\B=4

Now

\begin{aligned} &\frac{2 y}{(y+1)(2+y)}=\frac{-2}{y+1}+\frac{4}{y+2} \\ &I=-2 \int \frac{1}{y+1} d y+4 \int \frac{1}{y+2} d y \\ &I=-2 \log |y+1|+4 \log |y+2|+C \end{aligned}

\begin{aligned} &I=\log \left|\frac{(y+2)^{4}}{(y+1)^{2}}\right|+C \\ &I=\log \left|\frac{(\sin x+2)^{4}}{(\sin x+1)^{2}}\right|+C\quad\quad\quad\quad \quad[\because y=\sin x] \end{aligned}

 

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 extends single girl child scholarship registration deadline till November 20
anam-khan

CBSE reminds schools to submit registration data of Class 9, 11 students by October 16
anam-khan

CBSE reopens online portal for submission of LOC for Class 10, 12 board exams 2026; submit forms by Oct 8

Latest Question