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

Board Exam Result 2025 Live: UK Board 10th, 12th results tomorrow; AP SSC, TS Inter, UP, CBSE updates
anam-khan

When will CBSE Class 12 results 2025 be declared? Last two years' trends
anam-khan

CBSE Class 10, 12 board exams 2025 from April 7 to 11 for sports students

Latest Question