Evaluate the integral:

Key points to solve the problem:

- Such problems require the use of method of substitution along with method of integration by parts. By method of integration by parts if we have $\int \mathrm{f}(\mathrm{x}) \mathrm{g}(\mathrm{x}) \mathrm{dx}=\mathrm{f}(\mathrm{x}) \int \mathrm{g}(\mathrm{x}) \mathrm{dx}-\int \mathrm{f}^{\prime}(\mathrm{x})\left(\int \mathrm{g}(\mathrm{x}) \mathrm{dx}\right) \mathrm{dx}$

- To solve the integrals of the form: $\int \sqrt{a x^{2}+b x+c} d x$ after applying substitution and integration by parts we have direct formulae as described below:

$\int \sqrt{a^{2}-x^{2}} d x=\frac{x}{2} \sqrt{a^{2}-x^{2}}+\frac{a^{2}}{2} \sin ^{-1}\left(\frac{x}{a}\right)+C$

$\int \sqrt{x^{2}-a^{2}} d x=\frac{x}{2} \sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}-a^{2}}\right|+C$

$\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+C$

Let, $I=\int \sqrt{\left(2 x^{2}+3 x+4\right)} d x$

$\therefore I=\int \sqrt{2\left\{x^{2}+2\left(\frac{3}{4}\right) x+\left(\frac{3}{4}\right)^{2}+2-\left(\frac{3}{4}\right)^{2}\right\}} d x$

Using $a^{2}+2 a b+b^{2}=(a+b)^{2}$

We have:

$I=\sqrt{2} \int \sqrt{\left(x+\frac{3}{4}\right)^{2}+2-\frac{9}{16}} d x=\int \sqrt{\left(x+\frac{3}{4}\right)^{2}+\left(\frac{\sqrt{23}}{4}\right)^{2}} d x$

As I match with the form:

$\int \sqrt{x^{2}+a^{2}} d x=\frac{x}{2} \sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2} \log \left|x+\sqrt{x^{2}+a^{2}}\right|+C$

$\left.\therefore I={ }_{C} \sqrt{\frac{\left(x+\frac{3}{4}\right)}{2}} \sqrt{\left(x+\frac{3}{4}\right)^{2}+\left(\frac{\sqrt{23}}{4}\right)^{2}}+\frac{\left(\frac{\sqrt{23}}{4}\right)^{2}}{2} \log \left|\left(x+\frac{3}{4}\right)+\sqrt{\left(x+\frac{3}{4}\right)^{2}+\left(\frac{\sqrt{23}}{4}\right)^{2}}\right|\right\}+$

$\left.\Rightarrow 1=\frac{1}{8}(4 x+3) \sqrt{2\left\{\left(x+\frac{3}{4}\right)^{2}+\left(\frac{\sqrt{23}}{4}\right)^{2}\right.}\right\}+\frac{23 \sqrt{2}}{32} \log \left|\left(x+\frac{3}{4}\right)+\sqrt{\left(x+\frac{3}{4}\right)^{2}+\left(\frac{\sqrt{23}}{4}\right)^{2}}\right|+$ $\mathrm{C}$

$\Rightarrow I=\frac{1}{8}(4 x+3) \sqrt{2 x^{2}+3 x+4}+\frac{23 \sqrt{2}}{32} \log \left|\left(x+\frac{3}{4}\right)+\sqrt{x^{2}+\frac{3}{2} x+2}\right|+C$