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  • Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.21 Question 17 Maths Textbook Solution.

Explain Solution R.D.Sharma Class 12 Chapter 18 Indefinite Integrals Exercise 18.21 Question 17 Maths Textbook Solution.

Answers (1)

Answer: 5 \sqrt{x^{2}+4 x+10}-7 \log \left|x+2+\sqrt{x^{2}+4 x+10}\right|

Given: \int \frac{5 x+3}{\sqrt{x^{2}+4 x+10}} d x

Hint: Simplify the given function

Solution:

          \begin{aligned} &I=\int \frac{5 x+3}{\sqrt{x^{2}+4 x+10}} d x\\ &I=5 \int \frac{x+\frac{3}{5}}{\sqrt{x^{2}+4 x+10}} d x \text { multiplying and dividing by } 2 \text { in numerator, }\\ &I=\frac{5}{2} \int \frac{2 x+\frac{6}{5}}{\sqrt{x^{2}+4 x+10}} d x\\ &I=\frac{5}{2} \int \frac{2 x+4-\frac{14}{5}}{\sqrt{x^{2}+4 x+10}} d x \end{aligned}

         \begin{aligned} &I=\frac{5}{2} \int \frac{2 x+4}{\sqrt{x^{2}+4 x+10}} d x-7 \int \frac{1}{\sqrt{x^{2}+4 x+10}} d x \\ &I=\frac{5}{2} \int \frac{2 x+4}{\sqrt{x^{2}+4 x+10}} d x-7 \int \frac{1}{\sqrt{x^{2}+4 x+4+6}} d x \\ &I=\frac{5}{2} \int \frac{2 x+4}{\sqrt{x^{2}+4 x+10}} d x-7 \int \frac{1}{\sqrt{(x+2)^{2}+(\sqrt{6})^{2}}} d x \\ &I=\frac{5}{2} \int \frac{2 x+4}{\sqrt{x^{2}+4 x+10}} d x-7 \log \left|x+2+\sqrt{x^{2}+4 x+10}\right| \end{aligned}

                                                     using....................\left[\int \frac{1}{\sqrt{x^{2}+a^{2}}} d x=\log \left|x+\sqrt{x^{2}+a^{2}}\right|+c\right]

           Let

           \begin{aligned} &x^{2}+4 x+10=y \\ &(2 x+4) d x=d y \end{aligned}

   \Rightarrow \int \frac{2 x+4}{\sqrt{x^{2}+4 x+10}} d x=\int \frac{d y}{\sqrt{y}}=\frac{\sqrt{y}}{\frac{1}{2}}+c

                                                  \begin{aligned} &=2 \sqrt{y}+c \\ &=2 \sqrt{x^{2}+4 x+10}+c \end{aligned}

 I=5 \sqrt{x^{2}+4 x+10}-7 \log \left|x+2+\sqrt{x^{2}+4 x+10}\right|

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