Question

Evaluate : ($\large\textstyle\int(3x-2){\sqrt{x^{2}+x+1}}\,d x~.$

Find the approximate value of f(3·02), up to 2 places of decimal, where f(x) = 3x² + 5x + 3.

Find the particular solution of the differential equation : ${\frac{d y}{d x}}\ +y\cot x=2x+x^{2}\cot x,$ x ≠ 0. Given that y = 0, when $x={\frac{\pi}{2}}\ .$

Let f : W → W be defined as $f(n)={\binom{n+1,{\mathrm{if~}}~n{\mathrm{~is~even}}}{n-1,{\mathrm{~if~}}~n{\mathrm{~is~odd}}}}$ show that f is invertible. Find the inverse of f,where W is the set of all whole numbers.

A and B throw a pair of dice alternately. A wins the game if he gets a total of 9 and B wins if he gets a total of 7. If A starts the game, find the probability of winning the game by B.

Show that the relation S in the set R of real numbers defined as S = {(a, b) : a, b ∈ R and a ≤ b³} is neither reflexive nor symmetric nor transitive.

Solve the differential equation $x\log x\ {\frac{d y}{d x}}+y\ =$${\frac{2}{x}}\mathrm{log}\,x.$