Question

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

Find the angle of intersection of the curves x² + y² = 4 and (x – 2)² + y²= 4, at the point in the first quadrant.

Find the equations of the normal to the curve x² = 4y which passes through the point (1, 2).

Find the equations of the tangent and normal to the curve x = a sin³θ, y = b cos³θ at $\mathbf{\theta}={\frac{\pi}{4}}\,.$

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 the function f : R → R be given by f(x) = x² + 2 and g : R → R be given by g(x) = ${\frac{x}{x-1}},x\neq1,$ find fog and gof and hence find fog(2) and gof(– 3).

Find the particular solution of the following differential equation : $(x\ +1){\frac{d y}{d x}}=2e^{-y}-1;$ y = 0 when x = 0.