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Provide Solution For  R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise 18.20 Question 5 Maths Textbook Solution.

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Answer: -\frac{25}{3} \log |x+5|+\frac{4}{3} \log |x+2|+c

Given: \int \frac{x^{2}}{x^{2}+7 x+10} d x

Hint: Using Partial Fraction and  \int \frac{1}{x} d x

Explanation:

             Let I=\int \frac{x^{2}}{x^{2}+7 x+10} d x

            \frac{x^{2}}{x^{2}+7 x+10}=\frac{x^{2}}{x^{2}+5 x+2 x+10}=\frac{x^{2}}{x(x+5)+2(x+5)}=\frac{x^{2}}{(x+5)(x+2)}

           \frac{x^{2}}{(x+5)(x+2)}=\frac{A}{x+5}+\frac{B}{x+2}

            Multiply by  (x+5)(x+2)

            x^{2}=A(x+2)+B(x+5)

            Put x = -2

            4=A(0)+B(3) \Rightarrow B=\frac{4}{3}

            Put x = -5

           25=A(-3)+B(0) \Rightarrow A=-\frac{25}{3}

            \frac{x^{2}}{(x+5)(x+2)}=\frac{\frac{-25}{3}}{x+5}+\frac{\frac{4}{3}}{x+2}

            \therefore \int \frac{x^{2}}{(x+5)(x+2)} d x=\frac{-25}{3} \int \frac{1}{x+5} d x+\frac{4}{3} \int \frac{1}{x+2} d x

            =\frac{-25}{3} \log |x+5|+\frac{4}{3} \log |x+2|+c

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