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  • Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 1 Maths Textbook Solution.

Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise Revision Exercise Question 1 Maths Textbook Solution.

Answers (1)

Answer:

\frac{2}{3}\left[(x+1)^{\frac{3}{2}}-x^{\frac{8}{2}}\right]+c

Given:

\int \frac{1}{\sqrt{x}+(\sqrt{x+1})} d x

Hint:

Do Rationalization

Solution:   
we have,
\int \frac{1}{\sqrt{x}+(\sqrt{x+1})} d x

 =\int \frac{1}{\sqrt{x}+\sqrt{x+1}} \times \frac{\sqrt{x}-\sqrt{x+1}}{\sqrt{x}-\sqrt{x+1}} d x \quad(\because \text { rationalizing })

 \begin{aligned} &=\int \frac{\sqrt{x}-\sqrt{x+1}}{x-(x+1)} d x \ldots\left[(a-b)(a+b)=\left(a^{2}-b^{2}\right)\right] \\ &=\int-\frac{(\sqrt{x+1}-\sqrt{x)}}{-1} d x \end{aligned}

 \begin{aligned} &=\int \sqrt{x+1}-\sqrt{x} d x \\ &=\frac{(x+1)^{\frac{8}{2}}}{\frac{3}{2}}-\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+c \end{aligned}

 =\frac{2}{3}\left[(x+1)^{\frac{3}{2}}-x^{\frac{8}{2}}\right]+c

 

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