Correct option is: (3) \(1+3 \pi\)
Let, \(\mathrm{I}=\pi^{2} \int\limits_{-1}^{3 / 2}|\mathrm{x} \sin \pi \mathrm{x}| \mathrm{dx}\)
\(=\pi^{2}\left\{\int\limits_{-1}^{1} \mathrm{x} \sin \pi \mathrm{xdx}-\int\limits_{1}^{3 / 2} \mathrm{x} \sin \pi \mathrm{xdx}\right\}\)
\(=\pi^{2}\left\{2 \int\limits_{0}^{1} \mathrm{x} \sin \pi \mathrm{xdx}-\int\limits_{-1}^{3 / 2} \mathrm{x} \sin \pi \mathrm{xdx}\right\}\)
Consider
\(\int x \sin \pi x d x\)
\(-\mathrm{x} \cdot \frac{1}{\pi} \cos \pi \mathrm{x}+\int 1 \cdot \frac{1}{\pi} \cos \pi \mathrm{xdx}\)
\(=-\frac{x}{\pi} \cos \pi x+\frac{\sin \pi x}{\pi^{2}}\)
\(I=\pi^{2}\left\{2\left(-\frac{x}{\pi} \cos \pi x+\frac{\sin \pi x}{\pi^{2}}\right)_{0}^{1}-\left(-\frac{x}{\pi} \cos \pi x+\frac{\sin \pi x}{\pi^{2}}\right)_{1}^{3 / 2}\right\}\)
\(=\pi^{2}\left\{\frac{2}{\pi}-\left(-\frac{1}{\pi^{2}}-\frac{1}{\pi}\right)\right\}\)
\(=\pi^{2}\left\{\frac{3}{\pi}+\frac{1}{\pi^{2}}\right\}\)
\(=3 \pi+1\)
