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Find: $\int\frac{1}{x[(log~x)^{2}-3~log~x-4]}dx$

Evaluate : $\int_{-2}^{2}\sqrt{\frac{2-x}{2+x}}dx$

Show that: $\frac{d}{dx}(|x|)=\frac{x}{|x|},x\ne0$

If $x=e^{cos~3t}$ and $y=e^{sin~3t}$ , prove that $\frac{dy}{dx}=-\frac{y~log~x}{x~log~y}$

Show that $f(x)=e^{x}-e^{-x}+x-tan^{-1}x$ is strictly increasing in its domain.

Find: $\int\frac{e^{4x}-1}{e^{4x}+1}dx$

If M and m denote the local maximum and local minimum values of the function $f(x)=x+\frac{1}{x}(x\ne0)$ respectively, find the value of $(M-m)$

If $y=cosec(cot^{-1}x)$, then prove that $\sqrt{1+x^{2}}\frac{dy}{dx}-x=0$ .

Using integration, find the area of the region enclosed between the circle $x^{2}+y^{2}=16$ and the lines $x=-2$ and $x=2.$

A relation R on set $A=\{-4,-3,-2,-1,0,1,2,3,4\}$ be defined as $R=\{(x,y):x+y$ is an integer divisible by 2). Show that R is an equivalence relation. Also, write the equivalence class [2].

Solve the following linear programming problem graphically: Maximise $z=4x+3y.$ subject to the constraints $x+y\le800$, $2x+y\le1000$, $x\le400$, $x,y\ge0$.

Evaluate: $\int_{0}^{\pi/4}\frac{1}{sin~x+cos~x}dx$

Find: $\int\frac{2+sin~2x}{1+cos~2x}e^{x}dx$

Find: $\int\frac{x^{2}+1}{(x^{2}+2)(x^{2}+4)}dx$

Find the absolute maximum and absolute minimum values of the function f given by $f(x)=\frac{x}{2}+\frac{2}{x}$ , on the interval [1, 2].

Find the intervals in which the function $f(x)=\frac{log~x}{x}$ is strictly increasing or strictly decreasing.

Find the value of a and b so that function f defined as : $$ f(x) = \begin{cases} \frac{x-2}{|x-2|} + a, & \text{if } x < 2 \\ a+b, & \text{if } x = 2 \\ \frac{x-2}{|x-2|} + b, & \text{if } x > 2 \end{cases} $$ is a continuous function.

If $x~cos(p+y)+cos~p~sin(p+y)=0$ prove that $cos~p\frac{dy}{dx}=-cos^{2}(p+y),$ where p is a constant.

Let $\vec{a}$ and $\vec{b}$ be two non-zero vectors. Prove that $|\vec{a}\times\vec{b}|\le|\vec{a}||\vec{b}|$ . State the condition under which equality holds, i.e., $|\vec{a}\times\vec{b}|=|\vec{a}||\vec{b}|$

Find the position vector of point C which divides the line segment joining points A and B having position vectors $\hat{i}+2\hat{j}-\hat{k}$ and $-\hat{i}+\hat{j}+\hat{k}$ respectively in the ratio $4:1$ externally. Further, find $|\vec{AB}|:|\vec{BC}|$ .