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

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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

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Important Questions on Definite Integration

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

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

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

If

\int_{0}^{\frac{π}{2}} \log \cos x d x = \frac{π}{2} \log (\frac{1}{2}),

then

\int_{0}^{\frac{π}{2}} \log \sec x d x =

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

The integral value of

\int_{0}^{\frac{π}{2}} \frac{2^{\sin x}}{2^{\sin x} + 2^{\cos x}} d x =

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

The value of

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

is equal to

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

The integral

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

equals

EASY

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

\int_{- 2}^{2} |x| d x

is equal to

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

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

where

[x]

is greatest integer function

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

If

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-

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

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

is

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

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

EASY

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

If

\int_{- 1}^{4} f (x) d x = 4

and

\int_{2}^{4} (3 - f (x)) d x = 7

,

then

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

is

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

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

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

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

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

The integral value of

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

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

Let

x > 0

be a fixed real number. Then the integral

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

is equal to-

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

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)

is

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

Define a function

f : R \to R

by

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-

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

The value of the integral

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

a > 0

is

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

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

is

MEDIUM

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

The integral

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

is equal to