Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 44

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Given $f$ is an odd function defined everywhere, periodic with period $2$ and integrable on every interval. Let $g (x) = \int_{0}^{x} f (t) d t$ , then

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Important Questions on Definite Integration

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 45

Let

f : R \to R

be defined as

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

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 46

If

a, b \in \underset{}{R^{+}}

then

\lim_{n \to \infty} \sum_{k = 1}^{n} \frac{n}{(k + a n) (k + b n)}

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 47

Let

f (x) = \int_{x}^{x + \frac{π}{3}} |\sin θ| d θ (x \in [0, π])

.

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 48

If $f (x)$ is integrable over

[1, 2]

, then

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

is equal to

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 49

Let

I_{n} = \int_{0}^{\frac{1}{2}} \frac{1}{\sqrt{1 - x^{n}}} d x

where

n > 2

, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 50

If

f (x) = 2^{\{x\}}

, Where

\{x\}

denotes the fractional part of

x

. Then which of the following is true?

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 51

Let

f (x) = \int_{0}^{x} |2 t - 3| d t,

then

f

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 52

Let

I_{n} = \int_{0}^{π} {(\sin x)}^{n} d x, n \in N,

then