Question Number 53536 by gunawan last updated on 23/Jan/19
![If [x] stands for the gratest integer function the value of ∫_4 ^(10) (([x^2 ])/([x^2 −28x+196]+[x^2 ])) dx is](Q53536.png)
$$\mathrm{If}\:\left[{x}\right]\:\mathrm{stands}\:\mathrm{for}\:\mathrm{the}\:\mathrm{gratest}\:\mathrm{integer}\:\mathrm{function} \\ $$$$\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\int_{\mathrm{4}} ^{\mathrm{10}} \frac{\left[{x}^{\mathrm{2}} \right]}{\left[{x}^{\mathrm{2}} −\mathrm{28}{x}+\mathrm{196}\right]+\left[{x}^{\mathrm{2}} \right]}\:{dx}\:\mathrm{is} \\ $$$$ \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 23/Jan/19
![=using formula ∫_a ^b f(x)dx=∫_a ^b f(a+b−x)dx I=∫_4 ^(10) (([x^2 ])/([(14−x)^2 ]+[x^2 ]))dx =∫_4 ^(10) (([(14−x)^2 ])/([x^2 ]+[(14−x)^2 ]))dx 2I=∫_4 ^(10) dx 2I=∣x∣∣_4 ^(10) 2I=(10−4) I=3](Q53547.png)
$$={using}\:{formula}\:\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {f}\left({a}+{b}−{x}\right){dx} \\ $$$${I}=\int_{\mathrm{4}} ^{\mathrm{10}} \frac{\left[{x}^{\mathrm{2}} \right]}{\left[\left(\mathrm{14}−{x}\right)^{\mathrm{2}} \right]+\left[{x}^{\mathrm{2}} \right]}{dx} \\ $$$$=\int_{\mathrm{4}} ^{\mathrm{10}} \frac{\left[\left(\mathrm{14}−{x}\right)^{\mathrm{2}} \right]}{\left[{x}^{\mathrm{2}} \right]+\left[\left(\mathrm{14}−{x}\right)^{\mathrm{2}} \right]}{dx} \\ $$$$\mathrm{2}{I}=\int_{\mathrm{4}} ^{\mathrm{10}} {dx} \\ $$$$\mathrm{2}{I}=\mid{x}\mid\mid_{\mathrm{4}} ^{\mathrm{10}} \\ $$$$\mathrm{2}{I}=\left(\mathrm{10}−\mathrm{4}\right) \\ $$$${I}=\mathrm{3} \\ $$