Question

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

Using properties of determinants, prove the following : $\begin{vmatrix}a & a² & bc \\ b & b² & ca \\ c & c² & ab\end{vmatrix}$ = $(a-b)(b-c)(c-a)(b c+c a+a b).$

A ladder 5 m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 cm/s. How fast is its height on the wall decreasing, when the foot of the ladder is 4 m away from the wall ?

Find the equation of tangent to the curve 4x² + 9y² = 36 at the point (3 cos θ, 2 sin ).

Find the points on the curve y = x³ at which the slope of the tangent is equal to y-coordinate of the point.

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

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