Solve the following linear programming problem graphically : Maximise $\displaystyle Z=\mathrm{x}+2 \mathrm{y}$ Subject to the constraints : \[\begin{aligned} & \mathrm{x}-\mathrm{y} \geq 0 \\ & \mathrm{x}-2 \mathrm{y} \geq-2 \\ & \mathrm{x} \geq 0, \mathrm{y} \geq 0 \end{aligned} \]

Find : $\displaystyle \int \frac{\mathrm{x}+\sin \mathrm{x}}{1+\cos \mathrm{x}} \mathrm{~d} \mathrm{x}$

Evaluate: $\displaystyle \int_{0}^{\frac{\pi}{4}} \frac{\mathrm{~d} \mathrm{x}}{\cos ^{3} \mathrm{x} \sqrt{2 \sin 2 \mathrm{x}}}$

\[\begin{aligned} & \int \frac{\mathrm{x}+\sin \mathrm{x}}{1+\cos \mathrm{x}} d \mathrm{x} \\ & =\int \frac{\mathrm{x}+2 \sin \dfrac{\mathrm{x}}{2} \cos \dfrac{\mathrm{x}}{2}}{2 \cos ^{2} \dfrac{\mathrm{x}}{2}} d \mathrm{x} \\ & =\int \mathrm{x}\left(\frac{1}{2} \sec ^{2} \frac{\mathrm{x}}{2}\right) d \mathrm{x}+\int \tan \frac{\mathrm{x}}{2} d \mathrm{x} \\ & =\mathrm{x} \tan \frac{\mathrm{x}}{2}-\int \tan \frac{\mathrm{x}}{2} d \mathrm{x}+\int \tan \frac{\mathrm{x}}{2} d \mathrm{x} \\ & =\mathrm{x} \tan \frac{\mathrm{x}}{2}+C \end{aligned} \]

\[\begin{aligned} & \int_{0}^{\pi / 4} \frac{d \mathrm{x}}{\cos ^{3} \mathrm{x} \sqrt{2 \sin 2 \mathrm{x}}} \\ & =\frac{1}{2} \int_{0}^{\pi / 4} \frac{d \mathrm{x}}{\cos ^{4} \mathrm{x} \sqrt{\tan \mathrm{x}}} \\ & =\frac{1}{2} \int_{0}^{\pi / 4} \frac{\left(1+\tan ^{2} \mathrm{x}\right) \sec ^{2} \mathrm{x}}{\sqrt{\tan \mathrm{x}}} d \mathrm{x} \\ & \text { Put } \tan \mathrm{x}=t \Rightarrow \sec ^{2} \mathrm{x} d \mathrm{x}=d t \\ & \therefore I=\frac{1}{2} \int_{0}^{1} \frac{1+t^{2}}{\sqrt{t}} d t \\ & =\frac{1}{2} \int_{0}^{1}\left(\frac{1}{\sqrt{t}}+t^{3 / 2}\right) d t \\ & =\frac{1}{2}\left[2 \sqrt{t}+\frac{2}{5} t^{5 / 2}\right]_{0}^{1} \\ & =\frac{6}{5} \end{aligned} \]