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

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

Answers (1)

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

       

Posted by

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