Question

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

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

Using properties of determinants, prove that $\begin{vmatrix}1 & 1 & 1+3x \\ 1+3y & 1 & 1 \\ 1 & 1+3z & 1\end{vmatrix}$ = $9\bigl(3x y z+x y+y z+z x\bigr)$

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

Find the equation of tangent to the curve y = cos (x + y), -2π ≤ x ≤ 0, that is parallel to the line x + 2y = 0.

Find the general solution of the following differential equation : $y-x{\frac{d y}{d x}}=a\left(y^{2}+{\frac{d y}{d x}}\right).$

Evaluate : $\large\textstyle\int\!{\frac{\ d x}{x(x^{5}+3)}}\,.$