Question

Find : $\large\textstyle\int$$\large\frac{x^{2}\,d x}{x^{4}+x^{2}-12}\;.$

Let A = R – {2}, B = R – {1}. If f : A → B is a function defined by $f(x)\,=\,\left({\frac{x-1}{x-2}}\right)\!,$ show that f is one-one and onto. Hence find $f^{-1}.$

If y(x) is a solution of the differential equation $\left(\frac{\dot{2}+\sin x}{1+y}\right)\frac{d y}{d x}\;=-\cos x\;\mathrm{and}\;y\;(0)=1,$ then find the value of $y\left({\frac{\pi}{2}}\right).$

A bag contains (2n + 1) coins. It is known that (n – 1) of these coins have a head on both sides, whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is 31/42, determine the value of n.

Find the particular solution of the differential equation : $y e^{y}\,d x=(y^{3}+2x e^{y})d y,y(0)=1$

Using properties of determinants, prove that : $\begin{vmatrix}b+c & c+a & a+b \\ q+r & r+p & p+q \\ y+z & z+x & x+y\end{vmatrix}$ = 2 $\begin{vmatrix}a & b & c \\ p & q & r \\ x & y & z\end{vmatrix}$ .

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