Verify that the given function is a solution of the corresponding differential equation.

$y=c_1\sin x+c_2\cos x; \frac{d^{2}y}{d{x}^2}+y=0$

Verify that the given function is a solution of the corresponding differential equation.

$y\sec x=\tan x+c; \frac{dy}{dx}+y\tan x=\sec x$

Evaluate ${\int }_0^2\frac{1}{\sqrt{4-x^2}}dx$.

In a hockey match, both teams A and B scored same number of goals up to the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternately and decided that the team, whose captain gets a six first, will be declared the winner. If the captain of team A was asked to start, find their respective probabilities of winning the match and state whether the decision of the referee was fair or not.

Aman and Bhuvan throw a pair of dice alternately. In order to win, they have to get a sum of $8$. Find their respective probabilities of winning if Aman starts the game.

Evaluate the following:$\int \sqrt{\frac{{\cos }^{3}x}{{\sin }^{11}x}}dx$.

Evaluate the following:$\int \frac{1}{\sqrt{{\sin }^{3}x·\sin (x+\alpha )}}dx$.

Prove that

${\int }_0^{\frac{\pi }{2}}\frac{\cos x}{(\sin x+\cos x)}dx=\frac{\pi }{4}$