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Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.21 Question 12 maths Textbook Solution.

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Answer: 2 \sqrt{x^{2}+2 x+5}+3 \log \left|x+1+\sqrt{x^{2}+2 x+5}\right|+c

Given: \int \frac{2 x+5}{\sqrt{x^{2}+2 x+5}} d x

Hint: Simplify the given (f(x))n

Solution:

        \begin{aligned} &I=\int \frac{2 x+5}{\sqrt{x^{2}+2 x+5}} d x \\ &I=\int \frac{2 x+2+3}{\sqrt{x^{2}+2 x+5}} d x \\ &I=\int \frac{2 x+2}{\sqrt{x^{2}+2 x+5}} d x+3 \int \frac{1}{\sqrt{x^{2}+2 x+5}} d x \end{aligned}

       I=I_{1}+I_{2}                          .......................(1)

       Where

       I_{1}=\int \frac{2 x+2}{\sqrt{x^{2}+2 x+5}} d x \& I_{2}=3 \int \frac{1}{\sqrt{x^{2}+2 x+5}} d x

        Now,Let

        \begin{aligned} &x^{2}+2 x+5=y \\ &(2 x+2) d x=d y \end{aligned}                     ..........................(2)

        \begin{aligned} &I_{1}=2 \sqrt{y}+c \\ &I_{1}=2 \sqrt{x^{2}+2 x+5}+c \end{aligned}                                ( From equation 2)

       Now, I_{2}=3 \int \frac{1}{\sqrt{x^{2}+2 x+4+1}} d x

      I_{2}=3 \int \frac{1}{\sqrt{(x+1)^{2}+2^{2}}} d x                                                         \left[\int \frac{1}{\sqrt{x^{2}+a^{2}}} d x=\log \left|x+\sqrt{x^{2}+a^{2}}\right|+c\right]

      I_{2}=3 \log \left|x+1+\sqrt{x^{2}+2 x+5}\right|+c

      Putting value of I_{1} & I_{2}in equation (1)

      I=2 \sqrt{x^{2}+2 x+5}+3 \log \left|x+1+\sqrt{x^{2}+2 x+5}\right|+c

       

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