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 Indefinite Integrals exercise 18.16 question 12 maths

Explain solution RD Sharma class 12 chapter Indefinite Integrals exercise 18.16 question 12 maths

Answers (1)

Answer:

        \frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x^{2}-1}{\sqrt{3}}\right)+C

Hint:

Use substitution method as well as special integration formula to solve this type of problem

Given:

            \int \frac{x}{x^{4}-x^{2}+1}dx

Solution:

Let\: \: I=\int \frac{x}{x^{4}-x^{2}+1}dx

              =\int \frac{x}{(x^{2})^{2}-x^{2}+1}dx

Put\: \: x^{2}=t\Rightarrow 2x\, \, dx=dt\Rightarrow x\, dx=\frac{dt}{2}

            =\int \frac{1}{t^{2}-t+1}\, \frac{dt}{2}

            \begin{aligned} &=\frac{1}{2} \int \frac{1}{t^{2}-2 \cdot t \cdot \frac{1}{2}+\left(\frac{1}{2}\right)^{2}-\left(\frac{1}{2}\right)^{2}+1} d t \\ &=\frac{1}{2} \int \frac{1}{\left(t-\frac{1}{2}\right)^{2}-\frac{1}{4}+1} d t \\ &=\frac{1}{2} \int \frac{1}{\left(t-\frac{1}{2}\right)^{2}-\left(\frac{1-4}{4}\right)} d t \\ &=\frac{1}{2} \int \frac{1}{\left(t-\frac{1}{2}\right)^{2}-\left(\frac{-3}{4}\right)} d t \\ &=\frac{1}{2} \int \frac{1}{\left(t-\frac{1}{2}\right)^{2}+\frac{3}{4}} d t \end{aligned}

Let\: \: t-\frac{1}{2}=u\Rightarrow dt=du

Then \: \: I=\frac{1}{2} \int \frac{1}{u^{2}+\left(\frac{\sqrt{3}}{2}\right)^{2}} d u

                  \begin{aligned} &=\frac{1}{2} \times \frac{1}{\left(\frac{\sqrt{3}}{2}\right)} \tan ^{-1}\left(\frac{u}{\frac{\sqrt{3}}{2}}\right)+C \quad\left[\because \int \frac{1}{a^{2}+x^{2}} d x=\frac{1}{a} \tan ^{-1}\left(\frac{x}{a}\right)+C\right] \\ &=\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{t-\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)+C \quad\left[\because u=t-\frac{1}{2}\right] \\ &=\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{\frac{2 t-1}{2}}{\frac{\sqrt{3}}{2}}\right)+C \\ &=\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{2 t-1}{\sqrt{3}}\right)+C \\ &=\frac{1}{\sqrt{3}} \tan ^{-1}\left(\frac{2 x^{2}-1}{\sqrt{3}}\right)+C \quad\left[\because t=x^{2}\right] \end{aligned}

Posted by

Gurleen Kaur

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 issues final reminder to schools for supplementary exam LOC submission; submit by today without late fee
anam-khan

CBSE Supplementary Exam 2025: Registrations open for Class 10, 12 private students today
anam-khan

CBSE opens application to access 10th answer scripts; verification, revaluation for Class 12 starts on May 31

Latest Question