Let f n = ∫ 0 1 tan - 1 x n d x where n is a positive integer. Then

lim n → ∞ f n + f n + 2 exists and is equal to π 2

lim n → ∞ f n + f n + 2 exists and is equal to π

there exists a positive integer n 0 such that f n > f n + 1 for all positive integers n ≥ n 0

Let f n = ∫ 0 1 tan - 1 x n d x where n is a positive integer. Then

lim n → ∞ f n + f n + 2 exists and is equal to π 2

Define g x = ∫ - 3 3 f x - y f y d y , for all real x , where f t = 1 , 0 ≤ t ≤ 1 0 , e l s e w h e r e , then

g x is continuous everywhere and differentiable everywhere except at x = 0 , 1 , 2

g x is not continuous everywhere

g x is continuous everywhere but differentiable nowhere

g x is continuous everywhere and differentiable everywhere except at x = 0, 1

Statement - I : The value of the integral ∫ π / 6 π / 3 d x 1 + tan x is equal to π 6 . Statement - II : ∫ a b f x d x = ∫ a b f a + b - x d x .

Statement - I is false; Statement - II is true.

Statement - I is true; Statement - II is false.

Statement - I true; Statement - II is true; Statement - II is a correct explanation for Statement - I .

Statement - I is true; Statement - II is true; Statement - II is not a correct explanation for Statement - I .

HARD

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Let $f (n) = \int_{0}^{1} \tan^{- 1} (\sqrt[n]{x}) d x$ where $n$ is a positive integer. Then

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Important Questions on Definite Integration

MEDIUM

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

The value of

\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{x^{2} \cos x}{1 + e^{x}} d x

is equal to

MEDIUM

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

The value of the integral

\int_{- π}^{π} \frac{{c o s}^{2} x}{1 + a^{x}} d x,

a > 0

HARD

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

For a real number

x

, let

[x]

denote the largest integer less than or equal to

x

and

\{x\} = x - [x] .

Let

n

be a positive integer. Then

\int_{0}^{n} \cos (2 π [x] \{x\}) d x

is equal to

HARD

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

Define a function

f : R \to R

f (x) = \max \{|x|, |x - 1|, \dots ., |x - 2 n|\}

, where

n

is a fixed natural number. Then

\int_{0}^{2 n} f (x) d x

is-

HARD

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

Let

f : R \to R

be a function such that

f (2 - x) = f (2 + x)

and

f (4 - x) = f (4 + x)

, for all

x \in R

and

\int_{0}^{2} f (x) d x = 5

. Then the value of

\int_{10}^{50} f (x) d x

HARD

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

p (x)

is a cubic polynomial with

p (1) = 3, p (0) = 2

and

p (- 1) = 4

, then

\int_{- 1}^{1} p (x) d x

is-

EASY

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

\int_{0}^{3} [x] d x =______,

where

[x]

is greatest integer function

HARD

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

For a real number

x

let

[x]

denote the largest integer less than or equal to

x .

The smallest positive integer

n

for which the integral

\int_{1}^{n} [x] [\sqrt{x}] d x

exceeds

60

is-

HARD

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

The integral

\int_{0}^{π} \sqrt{1 + 4 {sin}^{2} \frac{x}{2} - 4 sin \frac{x}{2}} d x

equals

HARD

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

Let

f (x) = \max \{3, x^{2}, \frac{1}{x^{2}}\}

for

\frac{1}{2} \leq x \leq 2 .

Then the value of the integral

\int_{\frac{1}{2}}^{2} f (x) d x

HARD

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

For real

x

with

- 10 \leq x \leq 10

define

f (x) = \int_{- 10}^{x} 2^{[t]} d t

, where for a real number

r

we denote by

[r]

the largest integer less than or equal to

r

. The numbers of points of discontinuity of

f

in the interval

(- 10, 10)

EASY

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

Let

f

and

g

be continuous functions on

[0, a]

such that

f (x) = f (a - x)

and

g (x) + g (a - x) = 4

, then

\int_{0}^{a} f (x) g (x) d x

is equal to

HARD

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

For a real number

x

let

[x]

denote the largest integer less than or equal to

x a n d \{x\} - x - [x]

. The smallest possible integer value of n for which

\int_{1}^{n} [x] \{x\} d x

exceeds 2013 is

HARD

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

Let

x > 0

be a fixed real number. Then the integral

\int_{0}^{\infty} e^{- 1} |x - t| d t

is equal to-

HARD

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

Let

n

be a positive integer. For a real number

x,

let

[x]

denote the largest integer not exceeding

x

and

\{x\} = x - [x]

. Then

\int_{1}^{n + 1} \frac{{(\{x\})}^{[x]}}{[x]} d x

is equal to

HARD

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

The values of

\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{\sin^{2} x}{1 + 2^{x}} d x

HARD

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

The integral

\int_{0}^{\frac{1}{2}} \frac{\ln (1 + 2 x)}{1 + 4 x^{2}} d x

equals

HARD

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

Define

g (x) = \int_{- 3}^{3} f (x - y) f (y) d y,

for all real

x,

where

f (t) = \{\begin{matrix} 1, & 0 \leq t \leq 1 \\ 0, & e l s e w h e r e \end{matrix},

then

MEDIUM

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

Statement - I : The value of the integral $\int_{π / 6}^{π / 3} \frac{d x}{1 + \sqrt{\tan x}}$ is equal to $\frac{π}{6} .$

Statement - II : $\int_{a}^{b} f (x) d x = \int_{a}^{b} f (a + b - x) d x .$

MEDIUM

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

The integral

\int_{2}^{4} \frac{log x^{2}}{log x^{2} + log {(6 - x)}^{2}} d x

is equal to

Let fn=∫01tan-1xndx where n is a positive integer. Then

Important Questions on Definite Integration

Learn and Prepare for any exam you want !

Let $f (n) = \int_{0}^{1} \tan^{- 1} (\sqrt[n]{x}) d x$ where $n$ is a positive integer. Then