Question

Let $f\left(x\right)=e^{sinx}$ and $g(2-x)=x^2-f\left(x\right)$.
On the basis of above information, answer the following question :

If ${\int }_0^2\left(f\right(x)+g(x\left)\right)dx=k$ then $\left[k\right]$ is $($where $[.]$ denotes greatest integer function$)$

• A. $0$
• B. $2$
• C. $4$
• D. $5$

Answer 1

The correct option is B $2$

$g(2-x)=x^2-f\left(x\right)$

$\Rightarrow g(2-x)+f\left(x\right)=x^2.....................\left(1\right)$

Now,

${\int }_0^2\left(f\right(x)+g(x\left)\right)dx$

We know that ${\int }_b^{a}g\left(r\right)dr={\int }_b^{a}g(a+b-r)dr$

$={\int }_0^{2}f\left(x\right)dx+{\int }_0^{2}g\left(x\right)dx$

$={\int }_0^{2}f\left(x\right)dx+{\int }_0^{2}g(2-x)dx$

$={\int }_0^2\left(f\right(x)+g(2-x\left)\right)dx$

Using Equation (1)

$={\int }_0^2{x}^{2}dx=\frac{x^3}{3}{|}_0^2=\frac{8}{3}=k$

So, $\left[k\right]=2$