The integrating factor of the differential equation $x \frac{dy}{dx} + 2y = x^2 \log x$ is . . . . . . .

If the length of the sub-tangent at any point $P(x, y)$ on a curve $f(x, y) = 0$ is $x + 7y^2$, then $f(x, y) =$

If the solution of the differential equation $(1 + x^3) \frac{dy}{dx} + 6x^2y = 1 + x^2$ is $y = \frac{1}{(1 + x^3)^s} [x + \frac{x^p}{p} + \frac{x^q}{q} + \frac{x^r}{r} + c]$, then the $LCM$ of $p, q, r$ and $s$ is...

An integrating factor of the differential equation $x \frac{dy}{dx} + y \log x = x e^x x^{-\frac{1}{2} \log x}$,$(x > 0)$ is

If $x dy + (y + y^2 x) dx = 0$ and $y = 1$ at $x = 1$,then

The function $y=f(x)$ is the solution of the differential equation $\frac{dy}{dx}+\frac{xy}{x^2-1}=\frac{x^4+2x}{\sqrt{1-x^2}}$ in $(-1,1)$ satisfying $f(0)=0$. Then $\int_{-\frac{\sqrt{3}}{2}}^{\frac{\sqrt{3}}{2}} f(x) dx$ is