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

If [x] denotes the greatest integer less than or equal to x, then the value of the integral $${\int }_{-\pi /2}^{\pi /2}[[x]-\sin x]dx$$ is equal to :

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 $${\int }_0^{\theta }\sin xdx$$ 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?

If \int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R} where [x] is the greatest integer less than or equal to x, then the value of $$\alpha$$ is :

The value of the definite integral \int\limits_{\pi /24}^{5\pi /24} {{{dx} \over {1 + \root 3 \of {\tan 2x} }}} is :

Let $$f:[0,\infty )\to [0,\infty )$$ be defined as $$f(x)={\int }_0^x[y]dy$$ where [x] is the greatest integer less than or equal to x. Which of the following is true?

Let f : (a, b) $$\to$$ R be twice differentiable function such that $$f(x)={\int }_a^{x}g(t)dt$$ for a differentiable function g(x). If f(x) = 0 has exactly five distinct roots in (a, b), then g(x)g'(x) = 0 has at least :

If f(x) = \left\{ {\matrix{ {\int\limits_0^x {\left( {5 + \left| {1 - t} \right|} \right)dt,} } & {x > 2} \cr {5x + 1,} & {x \le 2} \cr } } \right., then

The value of the integral $$\underset{-1}{\overset{1}{\int }}\log \left(x+\sqrt{x^2+1}\right)dx$$ is :

The value of \mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{j = 1}^n {{{(2j - 1) + 8n} \over {(2j - 1) + 4n}}} is equal to :

The value of the definite integral \int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {{{dx} \over {(1 + {e^{x\cos x}})({{\sin }^4}x + {{\cos }^4}x)}}} is equal to :

The value of \int\limits_{{{ - 1} \over {\sqrt 2 }}}^{{1 \over {\sqrt 2 }}} {{{\left( {{{\left( {{{x + 1} \over {x - 1}}} \right)}^2} + {{\left( {{{x - 1} \over {x + 1}}} \right)}^2} - 2} \right)}^{{1 \over 2}}}dx} is :

The value of \mathop {\lim }\limits_{n \to \infty } {1 \over n}\sum\limits_{r = 0}^{2n - 1} {{{{n^2}} \over {{n^2} + 4{r^2}}}} is :

If the value of the integral \int\limits_0^5 {{{x + [x]} \over {{e^{x - [x]}}}}dx = \alpha {e^{ - 1}} + \beta }, where $$\alpha$$, $$\beta$$ $$\in$$ R, 5$$\alpha$$ + 6$$\beta$$ = 0, and [x] denotes the greatest integer less than or equal to x; then the value of ($$\alpha$$ + $$\beta$$)2 is equal to :

The value of \int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {\left( {{{1 + {{\sin }^2}x} \over {1 + {\pi ^{\sin x}}}}} \right)} \,dx is

If {U_n} = \left( {1 + {1 \over {{n^2}}}} \right)\left( {1 + {{{2^2}} \over {{n^2}}}} \right)^2.....\left( {1 + {{{n^2}} \over {{n^2}}}} \right)^n, then \mathop {\lim }\limits_{n \to \infty } {({U_n})^{{{ - 4} \over {{n^2}}}}} is equal to :

\int\limits_6^{16} {{{{{\log }_e}{x^2}} \over {{{\log }_e}{x^2} + {{\log }_e}({x^2} - 44x + 484)}}dx} is equal to :

The value of the integral \int\limits_0^1 {{{\sqrt x dx} \over {(1 + x)(1 + 3x)(3 + x)}}} is :

Let f be a non-negative function in [0, 1] and twice differentiable in (0, 1). If $${\int }_0^x\sqrt{1-{(f^{\prime }(t))}^2}dt={\int }_0^{x}f(t)dt$$, $$0\le x\le 1$$ and f(0) = 0, then \mathop {\lim }\limits_{x \to 0} {1 \over {{x^2}}}\int_0^x {f(t)dt} :