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 61

explain solution RD Sharma class 12 Chapter 18 Indefinite Integrals exercise 18.30 question 61

Answers (1)

Answer:

               \frac{1}{6} \log (\cos x-1)+\frac{1}{2} \log (\cos x+1)-\frac{2}{3} \log \left(\cos x+\frac{1}{2}\right)+C

Hint:

            To solve this integration, we use partial fraction method   

Given:

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

Explanation:

Let

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

Put

\begin{aligned} &\cos x=t \\ &-\sin x d x=d t \\ &d x=\frac{-1}{\sin x} d x \\ &I=\int \frac{-1}{\sin ^{2} x(1+2 t)} d t \\ &I=\int \frac{-1}{\left(1-\cos ^{2} x\right)(1+2 t)} d t \end{aligned}

I=\int \frac{1}{2\left(t^{2}-1\right)\left(\frac{1}{2}+t\right)} d t                            (1)

\frac{1}{(t-1)(t+1)\left(\frac{1}{2}+t\right)}=\frac{A}{t-1}+\frac{B}{t+1}+\frac{C}{\frac{1}{2}+t}                          (2)

1=A(t+1)\left(\frac{1}{2}+t\right)+B(t-1)\left(\frac{1}{2}+t\right)+C\left(t^{2}-1\right)

At t=- 1

\begin{aligned} &1=-2 B\left(\frac{-1}{2}\right) \\ &B=1 \end{aligned}

At t= 1

\begin{aligned} &1=2 A\left(\frac{3}{2}\right) \\ &A=\frac{1}{3} \end{aligned}

\begin{aligned} &\text { At } t=-\frac{1}{2} \\ &1=C\left(\frac{1}{4}-1\right) \\ &1=C\left(\frac{-3}{4}\right) \\ &C=\frac{-4}{3} \end{aligned}

Put in (1) using (2)
\begin{aligned} &I=\frac{1}{2} \cdot \frac{1}{3} \int \frac{1}{t-1} d t+\frac{1}{2} \int \frac{1}{t+1} d t-\frac{1}{2} \cdot \frac{4}{3} \int \frac{1}{t+\frac{1}{2}} d t \\ &I=\frac{1}{6} \log (t-1)+\frac{1}{2} \log (t+1)-\frac{2}{3} \log \left(t+\frac{1}{2}\right)+C \\ &I=\frac{1}{6} \log (\cos x-1)+\frac{1}{2} \log (\cos x+1)-\frac{2}{3} \log \left(\cos x+\frac{1}{2}\right)+C \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 10th, 12th results 2024 ‘shortly’; board issues Digilocker access code for students
anam-khan

CBSE Class 10, 12 results 2024 likely after May 20, says result portal
anam-khan

CBSE Result Date 2024 after May 20 at cbse.nic.in; Know Class 10th, 12th passing marks

Latest Question