Evaluate the following integrals:

Given $I=\int \frac{2 x+5}{\sqrt{x^{2}+2 x+5}} d x$

Integral is of form $\int \frac{\mathrm{px}+\mathrm{q}}{\sqrt{\mathrm{ax}^{2}+\mathrm{bx}+\mathrm{c}}} \mathrm{dx}$

Writing numerator as $p x+q=\lambda\left\{\frac{d}{d x}\left(a x^{2}+b x+c\right)\right\}+\mu$

$\Rightarrow p x+q=\lambda(2 a x+b)+\mu$

$\Rightarrow 2 x+5=\lambda(2 x+2)+\mu$

$\therefore \lambda=1$ and $\mu=3$

Let $2 x+5=2 x+2+3$ and split,

$\Rightarrow \int \frac{2 x+5}{\sqrt{x^{2}+2 x+5}} d x=\int\left(\frac{2 x+2}{\sqrt{x^{2}+2 x+5}}+\frac{3}{\sqrt{x^{2}+2 x+5}}\right) d x$

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

Consider $\int \frac{x+1}{\sqrt{x^{2}+2 x+5}} d x$

Let $u=x^{2}+2 x+5 \rightarrow d x=\frac{1}{2 x+2} d u$

$\Rightarrow \int \frac{\mathrm{x}+1}{\sqrt{\mathrm{x}^{2}+2 \mathrm{x}+5}} \mathrm{dx}=\int \frac{1}{2 \sqrt{\mathrm{u}}} \mathrm{du}$

$=\frac{1}{2} \int \frac{1}{\sqrt{\mathrm{u}}} \mathrm{du}$

We know that $\int x^{n} d x=\frac{x^{n+1}}{n+1}+c$

$\Rightarrow \frac{1}{2} \int \frac{1}{\sqrt{u}} \mathrm{du}=\frac{1}{2}(2 \sqrt{\mathrm{u}})$

$=\sqrt{\mathrm{u}}=\sqrt{\mathrm{x}^{2}+2 \mathrm{x}+5}$

Consider $\int \frac{1}{\sqrt{x^{2}+2 x+5}} d x$

$\Rightarrow \int \frac{1}{\sqrt{x^{2}+2 x+5}} d x=\int \frac{1}{\sqrt{(x+1)^{2}+4}} d x$

Let $\mathrm{u}=\frac{\mathrm{x}+1}{2} \rightarrow \mathrm{dx}=2 \mathrm{du}$

$\Rightarrow \int \frac{1}{\sqrt{(\mathrm{x}+1)^{2}+4}} \mathrm{dx}=\int \frac{2}{\sqrt{4 \mathrm{u}^{2}+4}} \mathrm{du}$

$=\int \frac{1}{\sqrt{\mathrm{u}^{2}+1}} \mathrm{du}$

We know that $\int \frac{1}{\sqrt{x^{2}+1}} d x=\sinh ^{-1} x+c$

$\Rightarrow \int \frac{1}{\sqrt{\mathrm{u}^{2}+1}} d \mathrm{u}=\sinh ^{-1}(\mathrm{u})$

$=\sinh ^{-1}\left(\frac{\mathrm{X}+1}{2}\right)$

Then,

$\Rightarrow \int \frac{2 x+5}{\sqrt{x^{2}+2 x+5}} d x=2 \int \frac{x+1}{\sqrt{x^{2}+2 x+5}} d x+3 \int \frac{1}{\sqrt{x^{2}+2 x+5}} d x$

$=2 \sqrt{x^{2}+2 x+5}+3 \sinh ^{-1}\left(\frac{x+1}{2}\right)+c$

$\therefore \mathrm{I}=\int \frac{2 \mathrm{x}+5}{\sqrt{\mathrm{x}^{2}+2 \mathrm{x}+5}} \mathrm{dx}=2 \sqrt{\mathrm{x}^{2}+2 \mathrm{x}+5}+3 \sinh ^{-1}\left(\frac{\mathrm{x}+1}{2}\right)+\mathrm{c}$