Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • School
  • Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 124 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise  Question 124 Maths Textbook Solution.

Answers (1)

Answer: \frac{1}{2} \log \left|\frac{1+x}{\sqrt{1+x^{2}}}\right|+\frac{1}{2} \tan ^{-1} x+C

Given:\int \frac{1}{1+x+x^{2}+x^{3}} d x

Hint: using partial function

Explanation: let I=\int \frac{1}{1+x+x^{2}+x^{2}} d x

\begin{aligned} &\frac{1}{1+x+x^{2}+x^{3}}=\frac{1}{(1+x)+x^{2}(1+x)}=\frac{1}{\left(1+x^{2}\right)(1+x)} \\ &\frac{1}{1+x+x^{2}+x^{3}}=\frac{1}{\left(1+x^{2}\right)(1+x)} \\ &\frac{1}{(1+x)\left(1+x^{2}\right)}=\frac{A}{(1+x)}+\frac{B x+C}{\left(1+x^{2}\right)} \end{aligned}

Multiplying by  (1+x)\left(1+x^{2}\right)

1=A\left(1+x^{2}\right)+(B x+C)(1+x)

\begin{aligned} &\text { put } x=-1 \\ &1=A(1+1)+(B(-1)+C)(1-1) \\ &1=A(2)+0=>A=\frac{1}{2} \end{aligned}

\begin{aligned} &\text { put } x=0 \\ &1=A(1+0)+(B(0)+C)(1+0) \\ &1=A+C=>1-\frac{1}{2}=C \Rightarrow \frac{1}{2} \end{aligned}

\begin{aligned} &\text { putting } x=1 \\ &1=A(1+1)+(B+C)(1+1) \\ &1=2 A+2 B+2 C \\ &1=2\left(\frac{1}{2}\right)+2 B+2\left(\frac{1}{2}\right) \\ &1=2 B+1=>2 B=1-2=>B=-\frac{1}{2} \end{aligned}

\begin{aligned} &\frac{1}{(1+x)\left(1+x^{2}\right)}=\frac{1}{2(1+x)}+\frac{1}{2}\left(\frac{-x+1}{\left(1+x^{2}\right)}\right) \\ &\int \frac{1}{(1+x)\left(1+x^{2}\right)} d x=\frac{1}{2} \int \frac{1}{1+x} d x-\frac{1}{2} \int \frac{x-1}{1+x^{2}} d x \\ &=\frac{1}{2} \int \frac{1}{1+x^{2}} d x-\frac{1}{2^{2}} \int \frac{2 x}{1+x^{2}} d x+\frac{1}{2} \int \frac{1}{1+x^{2}} d x \\ &=\frac{1}{2} \log |1+x|-\frac{1}{4} \log \left|1+x^{2}\right|+\frac{1}{2} \tan ^{-1} x+C \\ &=\frac{1}{2}\left[\log |1+x|-\log \left|\sqrt{1+x^{2}}\right|\right]+\frac{1}{2} \tan ^{-1} x+C \\ &=\frac{1}{2} \log \left|\frac{1+x}{\sqrt{1+x^{2}}}\right|+\frac{1}{2} \tan ^{-1} x+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

PM Modi's Pariksha Pe Charcha 2025 to be held next month, registrations open till January 14
anam-khan

CBSE Board Exams 2025: Online portal for availing exemptions by CWSN students opens
anam-khan

Board Exams: States agree to equivalence; no question paper ‘jumbling’ from next year, says PARAKH CEO

Latest Question