Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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Let $f$ be a non-negative function defined on the interval $[0,1]$ . If $\int_{0}^{x} \sqrt{1 - {(f^{'} (t))}^{2}} d t = \int_{0}^{x} f (t) d t, 0 \leq x \leq 1$ and $f (0) = 0$ , then

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Important Questions on Definite Integration

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

Let

f : [0, 2] \to R

be a function which is continuous on

[0, 2]

and is differentiable on

(0, 2) f (0) = 1

. Let

F (x) = \int_{0}^{x^{2}} f (\sqrt{t}) d t

for

x \in [0, 2] .

If

F^{'} (x) = f^{'} (x)

for all

x \in (0, 2),

then

F (2)

equals

EASY

Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

\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>Newton Leibnitz's Theorem

If the normal to the curve

y (x) = \int_{0}^{x} (2 t^{2} - 15 t + 10) d t

at a point

(a, b)

is parallel to the line

x + 3 y = - 5, a > 1

, then the value of

|a + 6 b|

is equal to ________.

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

If

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

, then

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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

EASY

Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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>Newton Leibnitz's Theorem

Let

F : [3, 5] \to R

be a twice differentiable function on

(3, 5)

such that

F (x) = e^{- x} \int_{3}^{x} (3 t^{2} + 2 t + 4 F^{'} (t)) d t .

If

F^{'} (4) = \frac{α e^{β} - 224}{{(e^{β} - 4)}^{2}}

, then

α + β

is equal to _____.

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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-

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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?

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

Let

f : R \to R

be defined as

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

If

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

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

If

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

then

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

is

EASY

Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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:

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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) .

If

f (x) = 0

has exactly five distinct roots in

(a, b),

then

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

has at least :

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

If

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

then the value of

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

is

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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

in

R

, then a possible value of

k

is

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

For

x \in R, x \neq 0,

if

y (x)

is a differentiable function such that

x \int_{1}^{x} y (t) d t = (x + 1) \int_{1}^{x} t y (t) d t,

then

y (x)

equals (where

C

is a constant)

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

Let

f : (- 1, 1) \to R

be a continuous function. If

\int_{0}^{\sin x} f (t) d t = \frac{\sqrt{3}}{2} x,

then

f (\frac{\sqrt{3}}{2})

is equal to:

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Mathematics>Integral Calculus>Definite Integration>Newton Leibnitz's Theorem

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