New chat PYQ Library Doubts Rank report

Sign in to see Recents

Sign in to save progress →

Definite Integration question

2014 · Shift 2 · Q38

Guest · try one question or challengeGuest · try one question or challenge

PYQ Library
JEE Advanced
Mathematics
Definite Integration
2014 · Shift 2 · Q38

Definite Integration question

2014 · Shift 2 · Q38

JEE AdvancedMathematicsDefinite IntegrationMCQ+3 / −1

Given that for each

a \in \left( {0,1} \right),\,\,\,\mathop {\lim }\limits_{h \to {0^ + }} \,\int\limits_h^{1 - h} {{t^{ - a}}{{\left( {1 - t} \right)}^{a - 1}}dt}

exists. Let this limit be

g(a).

In addition, it is given that the function

g(a)

is differentiable on

(0,1).

The value of

g'\left( {{1 \over 2}} \right)

is

A
${\pi \over 2}$
B
$\pi$
C
$-{\pi \over 2}$
D
$0$

More from Definite Integration

Given that for each $a \in (0, 1), h \to 0^{+} lim h \int 1 - h t^{- a} (1 - t)^{a - 1} d t$ exists. Let this limit be $g (a) .$ In addition, it is given that the…2014 · MCQ
Let $f$ $: [\frac{1}{2}, 1] \to R$ (the set of all real number) be a positive, non-constant and differentiable function such that $f^{'} (x) < 2 f (x)$ and $f (\frac{1}{2}) = 1.$ …2013 · MCQ
The value of the integral $- π /2 \int π /2 (x^{2} + 1 n \frac{π + x}{π - x}) cos x d x$ is2012 · MCQ
The value of $ℓ n 2 \int ℓ n 3 \frac{x s i n x ^{2}}{s i n x ^{2} + s i n ( ℓ n 6 - x ^{2} )} d x$ is2011 · MCQ
For any real number $x,$ let $[x]$ denote the largest integer less than or equal to $[- 10, 10]$ Let $f$ be a real valued function defined on the interval $[- 10, 10]$ by $f (x) = {x - [x] 1 + [x] - x i f [x] i s o dd, i f [x] i s e v e n$ …2010 · Numerical
The value of $0 \int 1 \frac{x ^{4} ( 1 - x ) ^{4}}{1 + x ^{2}} d x$ is (are)2010 · MCQ
The value of $x \to 0 lim \frac{1}{x ^{3}} 0 \int x \frac{t l n ( 1 + t )}{t ^{4} + 4} d t$ is2010 · MCQ
Let $f$ be a real-valued function defined on the interval $(- 1, 1)$ such that $e^{- x} f (x) = 2 + 0 \int x t^{4} + 1 d t,$ for all $x \in (- 1, 1)$ , and let $f^{- 1}$ be the inverse…2010 · MCQ