JEE Main 2006 — Calculus

If $x$ is real, the maximum value of $\frac{3 x^2+9 x+17}{3 x^2+9 x+7}$ is

If $x^m \cdot y^n=(x+y)^{m+n}$, then $\frac{d y}{d x}$ is

$\int_{-3 \pi / 2}^{-\pi / 2}\left[(x+\pi)^3+\cos ^2(x+3 \pi)\right] d x$ is equal to

$\int_0^\pi x f(\sin x) d x$ is equal to

The function $f(x)=\frac{x}{2}+\frac{2}{x}$ has a local minimum at

The set of points where $f(x)=\frac{x}{1+|x|}$ is differentiable is

The value of the integral, $\int_3^6 \frac{\sqrt{x}}{\sqrt{9-x}+\sqrt{x}} d x$ is

The value of $\int_1^a[x] f^{\prime}(x) d x, a>1$, where $[x]$ denotes the greatest integer not exceeding $x$ is