If the real part of the complex number $${(1 - \cos \theta + 2i\sin \theta )^{ - 1}}$$ is $${1 \over 5}$$ for $$\theta \in (0,\pi )$$, then the value of the integral $$\int_0^\theta {\sin x} dx$$ is equal to:

Let $$g(t) = \int_{ - \pi /2}^{\pi /2} {\cos \left( {{\pi \over 4}t + f(x)} \right)} dx$$, where $$f(x) = {\log _e}\left( {x + \sqrt {{x^2} + 1} } \right),x \in R$$. Then which one of the following is correct?

Let a be a positive real number such that $$\int_0^a {{e^{x - [x]}}} dx = 10e - 9$$ where [ x ] is the greatest integer less than or equal to x. Then a is equal to:

The value of the integral $$\int\limits_{ - 1}^1 {{{\log }_e}(\sqrt {1 - x} + \sqrt {1 + x} )dx} $$ is equal to: