Question

Find : $\large\textstyle\int$$\frac{x^{4}+1}{x\left(x^{2}+1\right)^{2}}d x$.

Let f : N → R be a function defined as f(x) = 4x² + 12x + 15. Then show that f : N → S, where S is range of f, is invertible. Also find the inverse of f.

Find the equations of the normal to the curve y = x³+ 2x + 6, which are parallel to line x + 14y + 4 = 0.

Solve the differential equation $x\log x\ {\frac{d y}{d x}}+y\ =$${\frac{2}{x}}\mathrm{log}\,x.$

Find the particular solution of the following differential equation : ${\frac{d y}{d x}}\;=1+x^{2}+y^{2}+x^{2}y^{2},$ given that y = 1, when x = 0.

Find the point on the curve y = x³ – 11x + 5 at which the equation of tangent is y = x – 11.

If A = $\left(\begin{array}{l l l }2 & 0 & 1 \\ 2 & 1 & 3 \\ 1 & -1 & 0\end{array}\right)$ , find A² – 5A + 4I and hence find a matrix X such that A² – 5A + 4I + X = 0.