Question

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

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

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

Using properties of determinants, prove that following : $\begin{vmatrix}a+x & y & z \\ x & a+y & z \\ x & y & a+z\end{vmatrix}$ = $a^{2}(a+x+y+z).$

Evaluate : $\large\textstyle\int(x-3){\sqrt{x^{2}+3x-18}}\,d x\,.$

Find the general solution of the differential equation (1 + tan y)(dx – dy) + 2xdy = 0.

Find the equation of tangent to curves x = sin 3t, y = cos 2t at $:t={\frac{\pi}{4}}\,.$