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

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)

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