(a) Evaluate: $$\int_\limits0^{2 \pi} \frac{1}{1+e^{\sin x}} d x$$

OR

$$\text { (b) Find: } \int \frac{x^4}{(x-1)\left(x^2+1\right)} d x$$

Find the area of the following region using integration:

$$\left\{(x, y): y^2 \leq 2 x \text { and } y \geq x-4\right\}$$

(a) Find the coordinates of the foot of the perpendicular drawn from the point $$P(0,2,3)$$ to the line $$\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}$$.

OR

(b) Three vectors $$\vec{a} \cdot \vec{b}$$ and $$\vec{c}$$ satisfy the condition $$\vec{a}+\vec{b}+\vec{c}=\overrightarrow{0}$$. Evaluate the quantity $$\mu= \vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a}$$, if $$|\vec{a}|=3,|\vec{b}|=4$$ and $$|\vec{c}|=2$$

Find the distance between the lines:

$$\begin{aligned} & \vec{r}=(\hat{i}+2 \hat{j}-4 \hat{k})+\lambda(2 \hat{i}+3 \hat{j}+6 \hat{k}) ; \\ & \vec{r}=(3 \hat{i}+3 \hat{j}-5 \hat{k})+\mu(4 \hat{i}+6 \hat{j}+12 \hat{k}) \end{aligned}$$