$\int e^x \left( \frac{2+\sin 2x}{1+\cos 2x} \right) dx$ is equal to

The angle between the lines $ab(x^2 - y^2) + (a^2 - b^2)xy = 0$ is

$\sum_{k=1}^{2n+1} (-1)^{k-1} k^2$ is equal to

$A$ metallic wire with tension $T$ and at temperature $30^{\circ} C$ vibrates with its fundamental frequency of $1 \ kHz$. The same wire with the same tension but at $10^{\circ} C$ temperature vibrates with a fundamental frequency of $1.001 \ kHz$. The coefficient of linear expansion of the wire is

If the angle $2 \theta$ is acute,then the acute angle between the pair of straight lines $x^2(\cos \theta - \sin \theta) + 2xy \cos \theta + y^2(\cos \theta + \sin \theta) = 0$ is:

$\sum_{k=1}^5 \frac{1^3+2^3+\ldots+k^3}{1+3+5+\ldots+(2k-1)}$ is equal to (in $.5$)