Mathematics - Definite Integration Question with Solution | TestHub

${\mathrm{f}}^{\prime }(\mathrm{x})=-\alpha \pi \sin (\pi \mathrm{x})$$\Rightarrow {\mathrm{f}}^{\prime }\left(\frac{1}{2}\right)=-\mathrm{a}\pi \sin \frac{\pi }{2}=-\mathrm{a}\pi =\pi \Rightarrow \mathrm{a}=-1$$\int (\mathrm{a}\cos \pi \mathrm{x}+\mathrm{b})\mathrm{dx}={\left(\frac{\mathrm{a}\sin \pi \mathrm{x}}{\pi }+\mathrm{bx}\right)}_{1/2}^{3/2}=\left(\frac{-\mathrm{a}}{\pi }+\frac{3\mathrm{b}}{2}\right)-\left(\frac{\mathrm{a}}{\pi }+\frac{\mathrm{b}}{2}\right)$$\Rightarrow \frac{2\mathrm{a}}{\pi }+\mathrm{b}=\frac{2}{\pi }+1\Rightarrow \mathrm{b}=1$ So, $\frac{-12}{\pi }\left({\sin }^{-1}(-1)+{\cos }^{-1}1\right)=\frac{-12}{\pi }\left(-\frac{\pi }{2}+0\right)=6$