Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 40

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JEE Advanced

IMPORTANT

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$\int_{0}^{3} \{\sqrt{x}\} d x,$ where $\{x\}$ denotes the fractional part of $x,$ is equal to

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Important Questions on Definite Integrals

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 41

If

I_{1} = \int_{0}^{a} [x] d x

and

I_{2} = \int_{0}^{a} \{x\} d x,

where

[x]

and

\{x\}

denote respectively, the integral and fractional parts of

x

and

a

is a positive integer, then

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 42

The value of

\int_{a}^{a + \frac{π}{2}} (|\sin x| + |\cos x|) d x

is

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 43

If

f (x)

is a periodic function with period

T,

then

\int_{a + 2 T}^{b + 2 T} f (x) d x

is equal to

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 44

\int_{0}^{\infty} \frac{x}{1 + e^{x}} d x

is equal to

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 45

The value of

\int_{e^{- 1}}^{e^{2}} |\frac{\log_{e} x}{x}| d x

is

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 46

If

\int_{1}^{2} e^{x^{2}} d x = a,

then the value of

\int_{e}^{e^{4}} \sqrt{\ln (x)} d x

is

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 47

For $x \in R$ and a continuous function $f,$ let

$I_{1} = \int_{\sin^{2} t}^{1 + \cos^{2} t} x f \{x (2 - x)\} d x$

and $I_{2} = \int_{\sin^{2} t}^{1 + \cos^{2} t} f \{x (2 - x)\} d x,$ then $\frac{I_{1}}{I_{2}}$ is

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JEE Advanced

IMPORTANT

Complete Study Pack for Engineering Entrances Objective Mathematics Vol 2>Chapter 10 - Definite Integrals>Target Exercises>Q 48

If

m = \int_{2}^{0} \frac{|\sin x|}{[\frac{x}{π}] + \frac{1}{2}} d x, n = \int_{0}^{2} \frac{|\sin x|}{[\frac{x}{π}] + \frac{1}{2}} d x,

where

[\cdot]

denotes greatest integer function, then