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

CBSE practical exams 2026 will be cancelled over non-compliance with guidelines, says board

Latest Question