Question

Find : $\textstyle\int$${\frac{x^{2}}{x^{4}+x^{2}-2}}d x$

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 $y\left({\frac{\pi}{2}}\right).$

Find the particular solution of the differential equation satisfying the given condition ${\frac{d y}{d x}}=y\tan x,$ given that y = 1, when x = 0.

Check whether the relation R in the set R of real numbers, defined by R = {(a, b) : 1 + ab > 0}, is reflexive, symmetric or transitive.

Find the particular solution of the differential equation $\frac{d y}{d x}\ \ +\ 2y\ \tan\ x\ =\ \sin\ x,$ given that y = 0 when $x={\frac{\pi}{3}}\,.$

To raise money for an orphanage, students of three schools A, B and C organized an exhibition in their locality, where they sold paper bags, scrap-books and pastel-sheets made by them using recycled paper at the rate of ₹ 20, ₹ 15 and ₹ 10 per unit respectively. School A sold 25 paper bags, 10 scrap-books and 30 pastel-sheets. School B sold 20 paper-bags, 15 scrap-books and 30 pastel-sheets. While school C sold 25 paper-bags, 18 scrap-books and 35 pastel-sheets. Using matrices, find the total amount raised by each school.

Solve the differential equation : $3e^{x}\tan y\,d x\ +(2-e^{x})\,\mathrm{sec}^{2}\,y\,d y=0,$ given that when x = 0, y = ${\frac{\pi}{4}}.$