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Need Solution for R.D.Sharma Maths Class 12 Chapter 18 Indefinite Integrals Exercise 18.21 Question 11 Maths Textbook Solution.

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

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

Hint: Simplify the given integration and solve it


           \begin{aligned} &I=\int \frac{x+1}{\sqrt{x^{2}+1}} d x \\ &I=\int \frac{x}{\sqrt{x^{2}+1}} d x+\int \frac{1}{\sqrt{x^{2}+1}} d x \end{aligned}

           \begin{aligned} &I=\frac{1}{2} \int \frac{2 x}{\sqrt{x^{2}+1}} d x+\log \left|x+\sqrt{x^{2}+1}\right|+c \\ &I=I_{1}+\log \left|x+\sqrt{x^{2}+1}\right|+c \\ &I_{1}=\frac{1}{2} \int \frac{2 x}{\sqrt{x^{2}+1}} d x \end{aligned}

           \begin{aligned} &\text { Let } \\ &x^{2}+1=y \\ &2 x d x=d y \end{aligned}

          \begin{aligned} &I_{1}=\frac{1}{2} \int \frac{d y}{\sqrt{y}}=\frac{1}{2}\left(\frac{\sqrt{y}}{\frac{1}{2}}\right)+c \\ &I_{1}=\sqrt{y}+c \\ &I_{1}=\sqrt{x^{2}+1}+c \\ &I=\sqrt{x^{2}+1}+\log \left|x+\sqrt{x^{2}+1}\right|+c \end{aligned}

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