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Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.3 Question 17 Maths Textbook Solution.

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Answer: \frac{2}{3}\left\{(x+3)^{\frac{3}{2}}+(x+2)^{\frac{3}{2}}\right\}+c

Hint: \text { To solve this equation we will add }(\mathrm{x}+3)+(\mathrm{x}+2 \text { ) in numerator and denominator }

Given: \int \frac{1}{\sqrt{x+3}-\sqrt{x+2}} d x

Solution: \int \frac{1}{\sqrt{x+3}-\sqrt{x+2}} d x

\begin{aligned} &=\int \frac{\sqrt{x+3}+\sqrt{x+2}}{(\sqrt{x+3}-\sqrt{x+2})(\sqrt{x+3}+\sqrt{x+2})} d x \\ &=\int \frac{\sqrt{x+3}+\sqrt{x+2}}{x+3-x-2} d x \\ &=\int \sqrt{x+3} d x+\int \sqrt{x+2} d x \end{aligned}

\begin{aligned} &{\left[\int(a x+b)^{n}=\frac{(a x+b)^{n+1}}{a(n+1)}+c, n \neq 1\right]} \\ &=\frac{2(x+3)^{\frac{3}{2}}}{3}+\frac{2(x+2)^{\frac{3}{2}}}{3}+c \\ &=\frac{2}{3}\left\{(x+3)^{\frac{3}{2}}+(x+2)^{\frac{3}{2}}\right\}+c \end{aligned}

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