Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 59

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Let $f (x)$ be a strictly increasing, non-negative function such that $f^{"} (x) < 0 \forall x \in R & I = \int_{α}^{β} f (x) d x (β > α)$ , 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 60

I_{1} = \int_{0}^{π} \frac{x \sin x}{1 + \cos^{2} x} d x, I_{2} = \int_{0}^{π} \frac{x^{3} \sin x}{(π^{2} - 3 π x + 3 x^{2}) (1 + \cos^{2} x)} d x,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 61

Let

L_{1} = \lim_{x \to 0^{+}} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x - \sin x}, L_{2} = \lim_{x \to 0^{-}} \frac{\int_{0}^{x^{2}} \sin \sqrt{t} d t}{x - \sin x},

then identify the correct option(s).

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 62

\lim_{n \to \infty} \frac{(1^{k} + 2^{k} + 3^{k} + \dots . . + n^{k})}{(1^{2} + 2^{2} + \dots . . + n^{2}) (1^{3} + 2^{3} + \dots \dots . + n^{3})} = F (k),

then

(k \in N)

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 63

Let

T_{n} = \sum_{r = 1}^{n} \frac{n}{r^{2} - 2 r . n + 2 n^{2}}, S_{n} = \sum_{r = 0}^{n - 1} \frac{n}{r^{2} - 2 r . n + 2 n^{2}},

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-2>Q 64

f (x) = \int_{0}^{1} f (t x) d t,

where

f^{'} (x)

is a continuous function such that

f (1) = 2,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-3>Q 1

If

I_{n} = \int_{- π}^{π} \frac{\sin n x}{(1 + π^{x}) \sin x} d x, n = 0, 1, 2, . . .,

then

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-3>Q 2

Let

f : R \to R

be a continuous function which satisfies

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

, then the value of

f (\ln 5)

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 30 - Definite Integration>Exercise-3>Q 3

\lim_{x \to 0} \frac{1}{x^{3}} \int_{0}^{x} \frac{t \ln (1 + t)}{t^{4} + 4} d t