31.

$\mathrm{Evaluate}: \int \frac{\sin \mathrm{x} + \cos \mathrm{x}}{16 + 9 \sin 2\mathrm{x}} \mathrm{dx}$

A factory manufactures two types of screws A and B, each type requiring the use of two machines, an automatic and a hand - operated. It takes 4 minutes on the automatic and 6 minutes on the hand-operated machines to manufacture a packet of screws ‘B’. Each machine is available for at most 4 hours on any day. The manufacturer can sell a packet of screws ‘A’ at a profit of 70 paise and screws ‘B’ at a profit of Rs. 1. Assuming that he can sell all the screws he manufactures, how many packets of each type should the factory owner produce in a day in order to maximize his profit? Formulate the above LPP and solve it graphically and find the maximum profit.

Evaluate:${\int }_1^3({\mathrm{x}}^2 + 3\mathrm{x} + {\mathrm{e}}^{\mathrm{x}}) \mathrm{dx}$

Using integration, find the area of the region in the first quadrant enclosed by the x-axis, the line y = x and the circle x² + y² =32

Find the distance of the point (-1,-51-10) from the point of intersection of the line $\overrightarrow{\mathrm{r}} = 2\hat{\mathrm{i}} -\hat{\mathrm{j}} + 2\hat{\mathrm{k}} + \mathrm{\lambda } (3\hat{\mathrm{i}} + 4\hat{\mathrm{j}} + 2\hat{\mathrm{k}})$ and the plane $\overrightarrow{r}. (\hat{i}-\hat{j} + \hat{k}) = 5$