Definite Integration

If I1 = \(\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{100}}} dx\) and
I2 = \(\int\limits_0^1 {{{\left( {1 - {x^{50}}} \right)}^{101}}} dx\) such that I2
= \(\alpha\)I1
then \(\alpha\)
equals to :

The integral \(\int\limits_1^2 {{e^x}.{x^x}\left( {2 + {{\log }_e}x} \right)} dx\) equals :

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

Let \(f(x) = \left| {x - 2} \right|\) and g(x) = f(f(x)), \(x \in \left[ {0,4} \right]\). Then \(\int\limits_0^3 {\left( {g(x) - f(x)} \right)} dx\) is equal to:

The integral \(\int\limits_{{\pi \over 6}}^{{\pi \over 3}} {{{\tan }^3}x.{{\sin }^2}3x\left( {2{{\sec }^2}x.{{\sin }^2}3x + 3\tan x.\sin 6x} \right)dx}\)
is equal to:

Let {x} and [x] denote the fractional part of x and the greatest integer \(\le\) x respectively of a real number x. If \(\int_0^n {\left\{ x \right\}dx} ,\int_0^n {\left[ x \right]dx}\) and 10(n2 – n), \(\left( {n \in N,n > 1} \right)\) ...

\(\int\limits_{ - \pi }^\pi {\left| {\pi - \left| x \right|} \right|dx}\) is equal to :

Suppose f(x) is a polynomial of degree four,
having critical points at –1, 0, 1. If
T = {x \(\in\) R |
f(x) = f(0)}, then the sum of squares of all the
elements of T is :

If the value of the integral
\(\int\limits_0^{{1 \over 2}} {{{{x^2}} \over {{{\left( {1 - {x^2}} \right)}^{{3 \over 2}}}}}} dx\)
is \({k \over 6}\), then k is equal to :

The integral \(\int\limits_0^2 {\left| {\left| {x - 1} \right| - x} \right|dx}\) is equal to______.

Let [t] denote the greatest integer less than or
equal to t. Then the value of \(\int\limits_1^2 {\left| {2x - \left[ {3x} \right]} \right|dx}\) is ______.

The value of \(\int\limits_0^\pi {{{\left| {\cos x} \right|}^3}} \,dx\) 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 :

Let f be a differentiable function from
R to R such that \(\left| {f\left( x \right) - f\left( y \right)} \right| \le 2{\left| {x - y} \right|^{{3 \over 2}}},\)    for all  \(x,y \in\) R.
If   \(f\left( 0 \right) = 1\)  
then   $$\int\...

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

The value of the integral \(\int\limits_0^1 {x{{\cot }^{ - 1}}(1 - {x^2} + {x^4})dx}\) 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 :-

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

Let \(f(x) = \int\limits_0^x {g(t)dt}\) where g is a non-zero even
function. If ƒ(x + 5) = g(x), then \(\int\limits_0^x {f(t)dt}\) equals-

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 \(\int\limits_0^a \,\)f(x) g(x) dx is equal to :

\(\mathop {\lim }\limits_{x \to \infty } \left( {{n \over {{n^2} + {1^2}}} + {n \over {{n^2} + {2^2}}} + {n \over {{n^2} + {3^2}}} + ..... + {1 \over {5n}}} \right)\) 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 : R \(\to\) R be a continuously differentiable function such that f(2) = 6 and f'(2) = \({1 \over {48}}\). If \(\int\limits_6^{f\left( x \right)} {4{t^3}} dt\) = (x - 2)g(x), then $$\mathop {\lim }\limits_{x \to 2} g\left( x \right)...

If \(\int\limits_0^{{\pi \over 2}} {{{\cot x} \over {\cot x + \cos ecx}}} dx\) = m(\(\pi\) + n), then m.n is equal to

A value of \(\alpha\) such that
\(\int\limits_\alpha ^{\alpha + 1} {{{dx} \over {\left( {x + \alpha } \right)\left( {x + \alpha + 1} \right)}}} = {\log _e}\left( {{9 \over 8}} \right)\) is :

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

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

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

If  \(\int\limits_0^x \,\)f(t) dt = x2 + \(\int\limits_x^1 \,\) t2f(t) dt then f '\(\left( {{1 \over 2}} \right)\) is -

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

The value of \(\int\limits_0^{2\pi } {\left[ {\sin 2x\left( {1 + \cos 3x} \right)} \right]} dx\),
where [t] denotes the greatest integer function is :

The integral \(\int\limits_{\pi /6}^{\pi /3} {{{\sec }^{2/3}}} x\cos e{c^{4/3}}xdx\) is equal to :

If \(f(x) = \int\limits_0^x {t\left( {\sin x - \sin t} \right)dt\,\,\,}\) then :

The value of the integral
\(\int\limits_{ - {\pi \over 2}}^{{\pi \over 2}} {{{\sin }^4}} x\left( {1 + \log \left( {{{2 + \sin x} \over {2 - \sin x}}} \right)} \right)dx\) is :

The value of integral \(\int_{{\pi \over 4}}^{{{3\pi } \over 4}} {{x \over {1 + \sin x}}dx}\) is :

If   \({I_1} = \int_0^1 {{e^{ - x}}} {\cos ^2}x{\mkern 1mu} dx;\)
   \({I_2} = \int_0^1 {{e^{ - {x^2}}}} {\cos ^2}x{\mkern 1mu} dx\)  and
\({I_3} = \int_0^1 {{e^{ - {x^3}}}} dx;\) then

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

If    \(\mathop {\lim }\limits_{n \to \infty } \,\,{{{1^a} + {2^a} + ...... + {n^a}} \over {{{(n + 1)}^{a - 1}}\left[ {\left( {na + 1} \right) + \left( {na + 2} \right) + ..... + \left( {na + n} \right)} \right]}} = {1 \over {60}}\)
for som...

If    \(\int\limits_1^2 {{{dx} \over {{{\left( {{x^2} - 2x + 4} \right)}^{{3 \over 2}}}}}} = {k \over {k + 5}},\) then k is equal to :

The integral \(\int_{{\pi \over {12}}}^{{\pi \over 4}} {\,\,{{8\cos 2x} \over {{{\left( {\tan x + \cot x} \right)}^3}}}} \,dx\) equals :

The integral \(\int\limits_{{\pi \over 4}}^{{{3\pi } \over 4}} {{{dx} \over {1 + \cos x}}}\) is equal to

If   \(2\int\limits_0^1 {{{\tan }^{ - 1}}xdx = \int\limits_0^1 {{{\cot }^{ - 1}}} } \left( {1 - x + {x^2}} \right)dx,\)
then \(\int\limits_0^1 {{{\tan }^{ - 1}}} \left( {1 - x + {x^2}} \right)dx\) is equalto :

The value of the integral
\(\int\limits_4^{10} {{{\left[ {{x^2}} \right]dx} \over {\left[ {{x^2} - 28x + 196} \right] + \left[ {{x^2}} \right]}}} ,\)
where [x] denotes the greatest integer less than or
equal to x, is :

For x \(\in\) R, x \(\ne\) 0, if y(x) is a differentiable function such that
x \(\int\limits_1^x y\) (t) dt = (x + 1) \(\int\limits_1^x ty\) (t) dt,  then y (x) equals :
(where C is a constant.)

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

The integral
\(\int\limits_2^4 {{{\log \,{x^2}} \over {\log {x^2} + \log \left( {36 - 12x + {x^2}} \right)}}dx}\) is equal to :

The integral \(\int\limits_0^\pi {\sqrt {1 + 4{{\sin }^2}{x \over 2} - 4\sin {x \over 2}{\mkern 1mu} } } dx\) equals:

Statement-1 : The value of the integral
\(\int\limits_{\pi /6}^{\pi /3} {{{dx} \over {1 + \sqrt {\tan \,x} }}}\) is equal to \(\pi /6\)
Statement-2 : $$\int\limits_a^b {f\left( x \right)} dx = \int\limits_a^b {f\left( {a + b - x} \right)}...

If \(g\left( x \right) = \int\limits_0^x {\cos 4t\,dt,}\) then \(g\left( {x + \pi } \right)\) equals