The integral $$\underset{\pi /6}{\overset{\pi /3}{\int }}{\sec }^{2/3}x\cos e{c}^{4/3}xdx$$ is equal to :

\mathop {\lim }\limits_{n \to \infty } \left( {{{{{(n + 1)}^{1/3}}} \over {{n^{4/3}}}} + {{{{(n + 2)}^{1/3}}} \over {{n^{4/3}}}} + ....... + {{{{(2n)}^{1/3}}} \over {{n^{4/3}}}}} \right) is equal to :

The value of $$\underset{0}{\overset{2\pi }{\int }}\left[\sin 2x\left(1+\cos 3x\right)\right]dx$$, where [t] denotes the greatest integer function is :

If f : R $$\to$$ R is a differentiable function and f(2) = 6, then \mathop {\lim }\limits_{x \to 2} {{\int\limits_6^{f\left( x \right)} {2tdt} } \over {\left( {x - 2} \right)}} is :-

The value of the integral $$\underset{0}{\overset{1}{\int }}x{\cot }^{-1}(1-x^2+x^4)dx$$ is :-

The value of \int\limits_0^{\pi /2} {{{{{\sin }^3}x} \over {\sin x + \cos x}}dx} is

Let $$f(x)=\underset{0}{\overset{x}{\int }}g(t)dt$$ where g is a non-zero even function. If ƒ(x + 5) = g(x), then $$\underset{0}{\overset{x}{\int }}f(t)dt$$ equals-

\mathop {\lim }\limits_{x \to \infty } \left( {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + {n \over {{n^2} + {3^2}}} + ..... + {1 \over {5n}}} \right) is equal to :

The integral \int\limits_1^e {\left\{ {{{\left( {{x \over e}} \right)}^{2x}} - {{\left( {{e \over x}} \right)}^x}} \right\}} \, loge x dx is equal to :

Let f and g be continuous functions on [0, a] such that f(x) = f(a – x) and g(x) + g(a – x) = 4, then $$\underset{0}{\overset{a}{\int }}$$f(x) g(x) dx is equal to :

The integral \int\limits_{\pi /6}^{\pi /4} {{{dx} \over {\sin 2x\left( {{{\tan }^5}x + {{\cot }^5}x} \right)}}} equals :

The value of the integral \int\limits_{ - 2}^2 {{{{{\sin }^2}x} \over { \left[ {{x \over \pi }} \right] + {1 \over 2}}}} \,dx (where [x] denotes the greatest integer less than or equal to x) is

If $$\underset{0}{\overset{x}{\int }}$$f(t) dt = x2 + $$\underset{x}{\overset{1}{\int }}$$ t2f(t) dt then f '\left( {{1 \over 2}} \right) is -

The value of \int\limits_{ - \pi /2}^{\pi /2} {{{dx} \over {\left[ x \right] + \left[ {\sin x} \right] + 4}}} , where [t] denotes the greatest integer less than or equal to t, is

Let $$\mathrm{I}=\underset{a}{\overset{b}{\int }}\left(x^4-2x^2\right)dx.$$ If I is minimum then the ordered pair (a, b) is -

If \int\limits_0^{{\pi \over 3}} {{{\tan \theta } \over {\sqrt {2k\,\sec \theta } }}} \,d\theta = 1 - {1 \over {\sqrt 2 }},\left( {k > 0} \right), then value of k is :

If f(x) = {{2 - x\cos x} \over {2 + x\cos x}} and g(x) = logex, (x > 0) then the value of integral \int\limits_{ - {\pi \over 4}}^{{\pi \over 4}} {g\left( {f\left( x \right)} \right)dx{\rm{ }}} is

The value of $$\underset{0}{\overset{\pi }{\int }}{\left|\cos x\right|}^{3}dx$$ is :

Let z $$\in$$ C with Im(z) = 10 and it satisfies {{2z - n} \over {2z + n}} = 2i - 1 for some natural number n. Then :

The equation |z – i| = |z – 1|, i = $$\sqrt{-1}$$, represents :