Let f : [0, π/2] → [0, 1] be the function defined by f(x) = sin² x and let g : [0, π/2] → [0, ∞) be the function defined by

$g\left(x\right)=\sqrt{\frac{\pi x}{2}-x^2}$

$\text{<br>}$

$\text{The value of}\mathrm{ \; 2}{\int }_0^{\frac{\pi }{2}}f\left(x\right)g\left(x\right)dx-{\int }_0^{\frac{\pi }{2}}g\left(x\right)dx\text{ is \_\_\_\_\_}$

The correct answer is 0.

Step 1: Understand the given integral expression

We are given the functions:

$f\left(x\right)={\sin }^{2}x$

$g\left(x\right)=\sqrt{\frac{\pi x}{2}-x^2}$

We need to find the value of the integral expression:

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

Step 2: Combine the integrals

By factoring out $g\left(x\right)$, we can express $I$ as a single integral:

$I={\int }_0^{\frac{\pi }{2}}\left(2f\right(x)-1)g\left(x\right)dx$

Substitute $f\left(x\right)={\sin }^{2}x$ into the integrand:

$2f\left(x\right)-1=2{\sin }^{2}x-1=-\cos \left(2x\right)$

Thus, the integral becomes:

$I={\int }_0^{\frac{\pi }{2}}-\cos \left(2x\right)g\left(x\right)dx$

Step 3: Apply King's Property of Definite Integrals

Using King's property, ${\int }_0^{a}h\left(x\right)dx={\int }_0^{a}h(a-x)dx$, where $a=\frac{\pi }{2}$:

First, evaluate gπ2-x:

$g\left(\frac{\pi }{2}-x\right)=\sqrt{\frac{\pi }{2}\left(\frac{\pi }{2}-x\right)-{\left(\frac{\pi }{2}-x\right)}^2}$

$=\sqrt{\frac{{\pi }^2}{4}-\frac{\pi x}{2}-\left(\frac{{\pi }^2}{4}-\pi x+x^2\right)}$

$=\sqrt{\frac{\pi x}{2}-x^2}=g\left(x\right)$

Next, evaluate $-\cos \left(2\left(\frac{\pi }{2}-x\right)\right)$:

$-\cos (\pi -2x)=-(-\cos (2x\left)\right)=\cos \left(2x\right)$

So, applying the substitution $x\to \frac{\pi }{2}-x$ gives:

$I={\int }_0^{\frac{\pi }{2}}\cos \left(2x\right)g\left(x\right)dx$

Step 4: Add the two expressions for I

Adding both forms of $I$ together:

$2I={\int }_0^{\frac{\pi }{2}}(-\cos (2x)+\cos (2x\left)\right)g\left(x\right)dx$

$2I={\int }_0^{\frac{\pi }{2}}0·g\left(x\right)dx=0$

$I=0$

Therefore, the value of the given expression is 0.