The value of the integral ∫ 4 10 x 2 x 2 − 28 x + 196 + x 2 d x , where x denotes the greatest…

Let $I= \underset{4}{\overset{10}{\int }}\frac{\left[x^2\right]}{\left[x^2-28x+196\right]+\left[x^2\right]}dx$

$\Rightarrow I= \underset{4}{\overset{10}{\int }}\frac{\left[x^2\right]}{\left[{\left(14-x\right)}^2\right]+\left[x^2\right]} dx ...\left(\mathrm{i}\right)$

Use $\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{a}{\overset{b}{\int }}f\left(a+b-x\right)dx$

$I= \underset{4}{\overset{10}{\int }}\frac{\left[{\left(14-x\right)}^2\right]}{\left[x^2\right]+\left[{\left(14-x\right)}^2\right]} dx ...\left(\mathrm{ii}\right)$ 

By adding equations $\left(\mathrm{i}\right) \& \left(\mathrm{ii}\right)$, we get

$2I= \underset{4}{\overset{10}{\int }}\frac{\left[{\left(14-x\right)}^2\right]+\left[x^2\right]}{\left[x^2\right]+\left[{\left(14-x\right)}^2\right]} dx$

$\Rightarrow 2I= \underset{4}{\overset{10}{\int }}dx$

$\Rightarrow 2I=6$

$\Rightarrow I=3$