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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.19 Question 8 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.19 Question 8 Maths Textbook Solution.

Answers (1)

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

Hint: You must know about how to solve integration

Given:\int \frac{2 x+5}{x^{2}-x-2} d x

Solution:

Let I=\int \frac{2 x+5}{x^{2}-x-2} d x

         =\int \frac{2 x-1+6}{x^{2}-x-2} d x

        =\int \frac{2 x-1}{x^{2}-x-2} d x+\int \frac{6}{x^{2}-x-2} d x

       I_{1}=\int \frac{2 x-1}{x^{2}-x-2} d x

Let

x^{2}-x-2=t

\Rightarrow(2 x-1) d x=d t

                        \begin{aligned} &I_{1}=\int \frac{d t}{t}=\log |t|+c_{1} \\ &=\log \left|x^{2}-x-2\right|+c_{1} \end{aligned}

          \begin{aligned} &I_{2}=\int \frac{6}{x^{2}-x-2} d x \\ &=6 \int \frac{d x}{x^{2}-x+\frac{1}{4}-\frac{1}{4}-2} \\ &=6 \int \frac{d x}{\left(x-\frac{1}{2}\right)^{2}-\left(\frac{3}{2}\right)^{2}}\left[\int \frac{d x}{x^{2}-a^{2}}=\frac{1}{2 a} \log \left|\frac{x-a}{x+a}\right|+c\right] \\ &=6 \times \frac{1}{3} \log \left|\frac{x-\frac{1}{2}-\frac{3}{2}}{x-\frac{1}{2}+\frac{3}{2}}\right|+c_{2} \\ &=2 \log \left|\frac{2 x-4}{2 x+2}\right|+c_{2} \end{aligned}

           So,I=I_{1}+I_{2}

           =\log \left|x^{2}-x-2\right|+2 \log \left|\frac{x-2}{x+1}\right|+c

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