Question

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

If C = 0·003x³ + 0·02x² + 6x + 250 gives the amount of carbon pollution in air in an area on the entry of x number of vehicles, then find the marginal carbon pollution in the air, when 3 vehicles have entered in the area.

If Z is the set of all integers and R is the relation on Z defined as R = {(a, b) : a, b ∈ Z and a – b is divisible by 5}. Prove that R is an equivalence relation.

Show that the relation R on the set Z of all integers defined by (x, y) ∈ $R\Leftrightarrow(x-y)$ is divisible by 3 is an equivalence relation.

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

Find : $\textstyle\int$${\frac{x^{2}}{x^{4}+x^{2}-2}}d x$

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