Let f : 0 , ∞ → R be a continuous function such that f x = 1 - 2 x + ∫ 0 x e x - t f t d t for all x ∈ 0 , ∞ . Then, which of the following statement(s) is (are) TRUE?

The area of the region x , y ∈ 0,1 × R : f x ≤ y ≤ 1 - x 2 is π - 2 4 Sq. unit

The curve y = f x pass through the point 1 , 2

The area of the region x , y ∈ 0,1 × R : f x ≤ y ≤ 1 - x 2 is π - 1 4 Sq. unit

HARD

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The value of $\int_{0}^{x} \frac{{(t - |t|)}^{2}}{1 + t^{2}}$ dt is equal to

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Important Questions on Integrals

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

Let

f : [0, \infty) \to R

be a continuous function such that

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

for all

x \in [0, \infty) .

Then, which of the following statement(s) is (are) TRUE?

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

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

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

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

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

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

, then

k

is equal to

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

Let

f (x) = 7 \tan^{8} x + 7 \tan^{6} x - 3 \tan^{4} x - 3 \tan^{2} x

for all

x \in (- \frac{π}{2}, \frac{π}{2}) .

Then the correct expression(s) is(are)

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

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:

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 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

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

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,

HARD

Mathematics>Integral Calculus>Integrals>Definite Integration

Let

F : R \to R

be a thrice differentiable function. Suppose that

F (1) = 0, F (3) = - 4

and

F^{'} (x) < 0

for all

x \in (\frac{1}{2}, 3) .

Let

f (x) = x F (x)

for all

x \in R

. If

\int_{1}^{3} x^{2} F^{'} (x) d x = - 12

and

\int_{1}^{3} x^{3} F^{''} (x) d x = 40,

then the correct expression(s) is(are)

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

Let

f^{'} (x) = \frac{192 x^{3}}{2 + \sin^{4} π x}

for all

x \in R

with

f (\frac{1}{2}) = 0 .

m \leq \int_{\frac{1}{2}}^{1} f (x) d x \leq M,

then the possible values of

m

and

M

are

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

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

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

The value of ∫ 0 xt – t 21+t2 dt is equal to

Important Questions on Integrals

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The value of $\int_{0}^{x} \frac{{(t - |t|)}^{2}}{1 + t^{2}}$ dt is equal to