HARD

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Let $f (x)$ be a differentiable non- decreasing function such that $\int_{0}^{x} {(f (t))}^{3} d t = \frac{1}{x^{2}} {(\int_{0}^{x} f (t) d t)}^{3} \forall x \in R - {0}$ and $f (1) = 1$ . If $\int_{0}^{x} f (t) d t = g (x)$ then $\frac{x g' (x)}{g (x)}$ is

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Important Questions on Integrals

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

The value of

\int_{- π / 2}^{π / 2} \frac{d x}{[x] + [\sin x] + 4},

where

[t]

denotes the greatest integer less than or equal to

t,

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

The value of

\int_{0}^{2} \frac{d x}{x^{2} - 2 x + 2}

is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

\int_{1}^{2} \frac{d x}{{(x^{2} - 2 x + 4)}^{\frac{3}{2}}} = \frac{k}{k + 5}

, then

k

is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

Suppose that the earth is a sphere of radius

6400

kilometers. The height from the earth's surface from where exactly a fourth of the earth's surface is visible, is

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral

\int_{1}^{e} \{{(\frac{x}{e})}^{2 x} - {(\frac{e}{x})}^{x}\} l o g_{e} x d x

is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

For

x, t \in R l e t p_{t} (x) = (\sin t) x^{2} - (2 \cos t) x + \sin t

be a family of quadratic polynomials in

x

with variable coefficients. Let

A (t) = \int_{0}^{1} p_{t} (x) d x

. Which of the following statement are true?

(I)

A (t) < 0

for all

t

.

(II)

A (t)

has infinitely many critical points.

(III)

A (t) = 0

for infinitely many

t

.

(IV)

A' (t) < 0

for all

t

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral

\int_{\frac{π}{12}}^{\frac{π}{4}} \frac{8 \cos 2 x}{{(\tan x + \cot x)}^{3}} d x

equals

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

I = \frac{2}{π} \int_{- \frac{π}{4}}^{\frac{π}{4}} \frac{d x}{(1 + e^{s i n x}) (2 - c o s 2 x)}

then

27 I^{2}

equals____

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

The number of continuous function

f : [0,1] \to [0,1]

such that

f (x) < x^{2}

for all

x

and

\int_{0}^{1} f (x) d x = \frac{1}{3}

is:

EASY

Mathematics>Integral Calculus>Integrals>Definite Integration

The integral

\int_{\frac{π}{4}}^{\frac{3 π}{4}} \frac{d x}{1 + \cos x}

is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

Let, the function

F

be defined as

F (x) = \int_{1}^{x} \frac{e^{t}}{t} d t, x > 0,

then the value of the integral

\int_{1}^{x} \frac{e^{t}}{t + a} d t,

where

a > 0,

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

For

x > 0,

let

f (x) = \int_{1}^{x} \frac{\log t}{1 + t} d t .

Then

f (x) + f (\frac{1}{x})

is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

2 \int_{0}^{1} \tan^{- 1} x d x = \int_{0}^{1} \cot^{- 1} (1 - x + x^{2}) d x

, then

\int_{0}^{1} \tan^{- 1} (1 - x + x^{2}) d x

is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

The value of the integral

\int_{0}^{\frac{1}{2}} \frac{1 + \sqrt{3}}{{({(x + 1)}^{2} {(1 - x)}^{6})}^{\frac{1}{4}}} d x

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

The value of the integral

\int_{0}^{1} x {c o t}^{- 1} (1 - x^{2} + x^{4}) d x

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

A value of

α

such that

\int_{α}^{α + 1} \frac{d x}{(x + α) (x + α + 1)} = l o g_{e} (\frac{9}{8})

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

\int_{0}^{\frac{π}{2}} \frac{c o t x}{c o t x + cosec x} d x = m (π + n),

then

m n

is equal to

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

\int_{0}^{1} x \tan^{- 1} x d x =

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

The number of continuous functions

f : [0, 1] \to R

that satisfy

\int_{0}^{1} x f (x) d x = \frac{1}{3} + \frac{1}{4} \int_{0}^{1} {(f (x))}^{2} d x

MEDIUM

Mathematics>Integral Calculus>Integrals>Definite Integration

\int_{0}^{π / 3} \frac{\tan θ}{\sqrt{2 k \sec θ}} d θ = 1 - \frac{1}{\sqrt{2}}, (k > 0)

, then the value of

k

Let f(x) be a differentiable non- decreasing function such that ∫0x(f(t))3dt=1x2∫0xftdt3∀x∈R−{0} and f(1)=1. If ∫0xf(t)dt=g(x) then xg'(x)g(x) is

Important Questions on Integrals

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Let $f (x)$ be a differentiable non- decreasing function such that $\int_{0}^{x} {(f (t))}^{3} d t = \frac{1}{x^{2}} {(\int_{0}^{x} f (t) d t)}^{3} \forall x \in R - {0}$ and $f (1) = 1$ . If $\int_{0}^{x} f (t) d t = g (x)$ then $\frac{x g' (x)}{g (x)}$ is