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Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.3 Question 5 Maths Textbook Solution.

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Answer: \frac{2}{3}\left\{(x+1)^{\frac{3}{2}}-x^{\frac{3}{2}}\right\}+c

Hint: \text { To solve this equation we will multiply } \sqrt{x+1}-\sqrt{x} \text { to numerator and denominator }

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

Solution: \int \frac{\sqrt{x+1}-\sqrt{x}}{(\sqrt{x+1}+\sqrt{x})(\sqrt{x+1}-\sqrt{x})}

\because \text { multiply } \sqrt{x+1}-\sqrt{x} \text { to numerator and denominator }

\begin{aligned} &=\int \frac{\sqrt{x+1}-\sqrt{x}}{x+1-x} d x \\ &=\int \sqrt{x+1} d x-\int \sqrt{x} d x \end{aligned}

                                                                                                                        \left[\int(a x+b)^{n}=\frac{(a x+b)^{n+1}}{a(n+1)}+c, n \neq c\right]

\begin{aligned} &=\frac{(x+1)^{\frac{1}{2}}+1}{\frac{1}{2}+1}-\frac{(x)^{\frac{1}{2}+1}}{\frac{1}{2}+1}+c \\ &=\frac{2}{3}\left\{(x+1)^{\frac{3}{2}}-x^{\frac{3}{2}}\right\}+c \end{aligned}

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