Question

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

Find : $\large\textstyle\int(3x+1)\large{\sqrt{4-3x-2x^{2}}}d x\,.$

Find the particular solution of the following differential equation : $x y{\frac{d y}{d x}}\,=(x+2)\,(y+2);y=-1,{\mathrm{when}}\,x=1.$

Find the particular solution of the differential equation $(1+e^{2x})d y+(1+y^{2})e^{x}d x=0,$ given that y = 1, when x = 0.

Prove that the function f : [0, ∞) → R given by f(x) = 9x² + 6x – 5 is not invertible. Modify the Codomain of the function f to make it invertible, and hence find f⁻¹.

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 $\begin{vmatrix}1 & 1 & 1+3x \\ 1+3y & 1 & 1 \\ 1 & 1+3z & 1\end{vmatrix}$ = $9\bigl(3x y z+x y+y z+z x\bigr)$