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Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.19 Question 6 Maths Textbook Solution.

Answers (1)

Answer: \frac{-4}{3} \ln |x-2|-\frac{2}{3} \ln |x+1|+c

Hint: You must know about how to solve integration

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

Solution: \int \frac{2 x}{2+x-x^{2}} d x

                =-\int \frac{2 x}{x^{2}-x-2} d x

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

\frac{A}{(x-2)}+\frac{B}{(x+1)}=\frac{2 x}{(x-2)(x+1)}

A(x+1)+B(x+2)=2 x

A x+A+B x+2 B=2 x

On comparing,

                    \left(\begin{array}{c|c} A x+B x=2 x & A-2 B=0 \\ A+B=2 & A=2 B \end{array}\right)

                      2 B+B=2

                      B=\frac{2}{3}, A=\frac{4}{3}

                    \int-\frac{4}{3}\left(\frac{1}{x-2}\right) d x-\frac{2}{3} \int \frac{1}{x+1} d x

                    =-\frac{4}{3} \log |x-2|-\frac{2}{3} \log |x+1|+c


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