First slide

Methods of integration

Question

Let $F (x) = \int e^{- x^{2}} x^{5} d x$
Statement-1: If $F (0) = 0$ , then $F (1)$ $= 1 - \frac{5}{2} e^{- 1}$
Statement-2: $F$ increases on (0, $\infty$ )

Moderate

A

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
B

STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for
STATEMENT-1
C

STATEMENT-1 is True, STATEMENT-2 is False
D

STATEMENT-1 is False, STATEMENT-2 is True

Solution

Put $x^{2} = t, \int e^{- x^{2}} x^{5} d x = \frac{1}{2} \int e^{- t} t^{2} d t$
$= C - \frac{1}{2} e^{- x^{2}} (x^{4} + 2 x^{2} + 2)$
$F (0) = 0 \Rightarrow C = 1$
Hence $F (x) = 1 - \frac{1}{2} e^{- x^{2}} (x^{4} + 2 x^{2} + 2)$
$F^{'} (x) = e^{- x^{2}} x^{5} > 0 x \in (0, \infty)$

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