Let f : R → R be a continuous function satisfying f x + ∫ 0 x t f t d t + x 2 = 0 . For all x ∈ R . Then-

f x has more than one point in common with the x -axis

A function f is continuous for all x (and not everywhere zero) such that f 2 x = ∫ 0 x f t cost 2 + sin t d t then f x is:

1 2 log e 2 + sin x 2 ; x ≠ n π , n ∈ I

1 2 log e x + cos x 2 ; x ≠ 0

1 2 log e 3 2 + cos x ; x ≠ 0

cos x + sin x 2 + sin x ; x ≠ n π + 3 π 4 , n ∈ I

If f x = ∫ 0 x t sin x - sin t d t , then

f ''' x + f ' x = cos x - 2 x sin x

f ''' x - f '' x = cos x - 2 x sin x

f ''' x + f '' x - f ' x = cos x

HARD

Earn 100

In $(- 4, 4)$ the function $f (x) = \int_{- 10}^{x} (t^{4} - 4) e^{- 4 t} d t$ has

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Important Questions on Definite Integration

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

The least value of the function $f (x) = \int_{0}^{x} (3 \sin x + 4 \cos x) d x$ in the interval $[\frac{5 π}{4}, \frac{4 π}{3}]$ is

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

Let a function $f : [0, 5] \to R$ be continuous, $f (1) = 3$ and $F$ be defined as: $F (x) = \int_{1}^{x} t^{2} g (t) d t,$ where $g (t) = \int_{1}^{t} f (u) d u .$ Then for the function $F (x),$ the point $x = 1$ is:

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

\lim_{x \to 1} (\frac{\int_{0}^{(x - 1)^{2}} t \cos t^{2} d t}{(x - 1) \sin (x - 1)})

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

Suppose a continuous function

f : (0, \infty) \to R

satisfies

f (x) = 2 \int_{0}^{x} t f (t) d t + 1, \forall x \geq 0

. Then,

f (1)

equals

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

Let

f : R \to R

be a continuous function satisfying

f (x) + \int_{0}^{x} t f (t) d t + x^{2} = 0 .

For all

x \in R .

Then-

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

The value of

\lim_{x \to 0} \frac{1}{x} [\int_{y}^{a} e^{\sin^{2} t} d t - \int_{x + y}^{a} e^{\sin^{2} t} d t]

is equal to

EASY

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

\underset{x \to 0}{l i m} \frac{\int_{0}^{x} t s i n (10 t) d t}{x}

, is equal to

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

Let

f : (a, b) \to R

be twice differentiable function such that

f (x) = \int_{a}^{x} g (t) d t

for a differentiable function

g (x) .

f (x) = 0

has exactly five distinct roots in

(a, b),

then

g (x) g^{'} (x) = 0

has at least :

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

A continuous function

f : R \to R

satisfies equation

f (x) = x + \int_{0}^{x} f (t) d t .

Which of the following options is true?

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

f : R \to R

is a differentiable function and

f (2) = 6,

then

\underset{x \to 2}{l i m} \int_{6}^{f (x)} \frac{2 t d t}{(x - 2)}

is:

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

\lim_{x \to 0} \frac{\int_{0}^{x^{2}} (\sin \sqrt{t}) d t}{x^{3}}

is equal to:

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

The limit

\underset{x \to \infty}{l i m} x^{2} \int_{0}^{x} e^{t^{3} - x^{3}} d t

equals

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

Let

f : R \to R

be defined as

f (x) = e^{- x} \sin x .

F : [0, 1] \to R

is a differentiable function such that

F (x) = \int_{0}^{x} f (t) d t,

then the value of

\int_{0}^{1} (F^{'} (x) + f (x)) e^{x} d x

lies in the interval

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

The intercepts on the

x

-axis made by tangents to the curve,

y = \int_{0}^{x} |t| d t, x \in R,

which are parallel to the line

y = 2 x

, are equal to

EASY

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

A function

f

is continuous for all

x

(and not everywhere zero) such that

f^{2} (x) = \int_{0}^{x} f (t) \frac{cost}{2 + \sin t} d t

then

f (x)

is:

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

f (x) = \int_{e}^{e^{x}} \log_{e} (\frac{x}{\log_{e} t}) d t,

then the value of

\frac{3 f^{'} (3)}{e}

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

Let

F (x) = \int_{0}^{x} f (t) d t

, where

f (x) = 2 + \sin x - \cos x

. If

|F (x) - F (y)| \leq k |x - y|

for all

x

and

y

R

, then a possible value of

k

EASY

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

Let

f : R \to R

be a continuous and differentiable function such that

f (2) = 6

and

f^{'} (2) = \frac{1}{48}

. If

\int_{6}^{f (x)} 4 t^{3} d t = (x - 2) g (x),

then

\underset{x \to 2}{l i m} g (x)

is equal to

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

f (x) = \int_{0}^{x} t (\sin x - \sin t) d t

, then

HARD

Mathematics>Integral Calculus>Definite Integration>Properties of Definite Integrals

\int_{0}^{x} f (t) d t = x^{2} + \int_{x}^{1} t^{2} f (t) d t,

then

f^{'} (\frac{1}{2})

In (-4, 4) the function fx=∫-10x(t4-4)e-4tdt has

Important Questions on Definite Integration

Learn and Prepare for any exam you want !

In $(- 4, 4)$ the function $f (x) = \int_{- 10}^{x} (t^{4} - 4) e^{- 4 t} d t$ has