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Find: $\int\frac{x-\sin x}{1-\cos x}dx$

Find: $\int\frac{x+3}{x^{2}+4x+5}dx$

Find: $\int\frac{\cos x}{(2+\sin x)(4+\sin x)}dx$

Find: $\int\tan^{-1}(\frac{1-x}{1+x})dx$

Find: $\int\frac{dx}{x^{1/2}+x^{1/3}}$

If $\frac{d}{dx}(F(x))=\frac{1}{e^{x}+1}$ then find $F(x)$ given that $F(0)=\log\frac{1}{2}$

Find: $\int\frac{x+2}{\sqrt{9x-x^{2}}}dx$

A spherical balloon loses its volume due to escape of air from it in such a way that decrease of volume at any instant is proportional to its surface area. Show that the radius is decreasing at a constant rate.

Differentiate $\tan^{-1}(\frac{\sqrt{1+x^{2}}-\sqrt{1-x^{2}}}{\sqrt{1+x^{2}}+\sqrt{1-x^{2}}})$ with respect to $\cos^{-1}x^{2}$.

If $xy=e^{x-y}$, then find $\frac{dy}{dx}$.

If $(\sin x)^{y}=y^{\cos x}$ then find $\frac{dy}{dx}$.

If $A=\begin{bmatrix}3&2&0\\1&4&0\\0&0&5\end{bmatrix}$, then compute $A^{2}-7A+10~I$.

Find the angle between the following pair of lines: $\frac{x-2}{3}=\frac{y+5}{2}=\frac{1-z}{-6}$ and $\frac{x-7}{1}=\frac{y}{2}=\frac{6-z}{-2}$.

Find the vector equation of a line passing through the origin and perpendicular to both the lines $\vec{r}=2\hat{i}-\hat{j}+2\hat{k}+\lambda(3\hat{i}+4\hat{j}+2\hat{k})$ and $\vec{r}=\mu(\hat{i}-\hat{j}+\hat{k})$.

If the lines $\frac{x-3}{1}=\frac{1-y}{1}=\frac{z+2}{p}$ and $\frac{2-x}{3}=\frac{y+1}{5}=\frac{z+56}{2p}$ are perpendicular to each other, then find the value(s) of p.

Find the co-ordinates of the point on the line $\vec{r}=-\hat{j}+3\hat{k}+\lambda(2\hat{i}-2\hat{j}+\hat{k})$ such that the sum of co-ordinates is 3.

If A, B and C be three non-collinear points such that $\vec{AB}=\hat{i}+2\hat{j}-\hat{k}$ and $\vec{AC}=2\hat{i}-3\hat{j}$, then find the area of $\Delta ABC$.

Three honey bees were found flying along the vectors $\vec{a}=2\hat{i}-3\hat{j}+\hat{k}$, $\vec{b}=4\hat{j}-2\hat{k}$ and $\vec{c}=3\hat{i}+2\hat{k}$ respectively. Find the value of $\lambda$ such that the path for $\vec{a}+\lambda\vec{b}$ is perpendicular to $\vec{c}$.

A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with x-axis, $\frac{\pi}{3}$ with y-axis and an acute angle $\theta$ with z-axis. Find $\theta$ and the components of $\vec{a}$.

Let two rods placed on the ground be represented by vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$. Find a vector representing a flag-post of height 5 m that has to be erected perpendicular to both the rods.