$\text{ The value of the definite integral }$${\int }_0^{3\pi /4}\left[\right(1+x)\sin x+(1-x\left)\cos x\right]dx\text{ is }-$

$\mathrm{I}={\int }_0^{3\pi /4}(\mathrm{sinx}+\mathrm{cosx})\mathrm{dx}+{\int }_0^{3\pi /4}\underset{\mathrm{I}}{\underbrace{\mathrm{x}}}\left(\underset{\mathrm{II}}{\underbrace{\mathrm{sinx}-\mathrm{cosx}}}\right)\mathrm{dx}$

$={\int }_0^{3\pi /4}(\sin x+\cos x)dx+\underset{\text{zero }}{\underbrace{{\left.x(-\cos x-\sin x)\right|}_0^{3\pi /4}}}$$+{\int }_0^{3\pi /4}(\sin x+\cos x)dx$

$=2{\int }_0^{3\pi /4}(\sin x+\cos x)dx=2(\sqrt{2}+1)=2\tan \frac{3\mathrm{\pi }}{8}$