JEE Main 2007 — Calculus

$\int \frac{d x}{\cos x+\sqrt{3} \sin x}$ equals

The solution for $x$ of the equation $\int_{\sqrt{2}}^x \frac{d t}{t \sqrt{t^2-1}}=\frac{\pi}{2}$ is

The function $f(x)=\tan ^{-1}(\sin x+\cos x)$ is an increasing function in

Let $F(x)=f(x)+f\left(\frac{1}{x}\right)$, where $f(x)=\int_1^x \frac{\log t}{1+t} d t$. Then $F(e)$ equals

Let $f: R \rightarrow R$ be a function defined by $f(x)=\operatorname{Min}\{x+1,|x|+1\}$. Then which of the following is true?

The function $f:R \sim\{0\} \rightarrow R$ given by $f(x)=\frac{1}{x}-\frac{2}{e^{2 x}-1}$ can be made continuous at $x=0$ by defining $f(0)$ as

The area enclosed between the curves $y^2=x$ and $y=|x|$ is