If $I = \int_{0}^{1} e^{x^{2}} d x$ , then which of the following is correct ?

$(a) I < 1 (b) I \leq 0$
$(c) I > 1 (d) I \leq 1$

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Solution

$e > 1 ∴ \forall x \in (0, 1), e^{x} > 1 ∴ e^{x^{2}} > 1$

Taking integration both the sides
$\Rightarrow 1 \int 0 e^{x^{2}} d x > 1 \int 0 d x \Rightarrow I > 1$

Option $(c)$ is correct.

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If I=∫10ex2dx, then which of the following is correct ? (a) I<1 (b) I≤0 (c) I>1 (d) I≤1

e>1∴∀x∈(0,1),ex>1∴ex2>1 Taking integration both the sides ⇒1∫0ex2dx>1∫0dx⇒I>1 Option (c) is correct.

If $I = \int_{0}^{1} e^{x^{2}} d x$ , then which of the following is correct ?

$(a) I < 1 (b) I \leq 0$
$(c) I > 1 (d) I \leq 1$

$e > 1 ∴ \forall x \in (0, 1), e^{x} > 1 ∴ e^{x^{2}} > 1$

Taking integration both the sides
$\Rightarrow 1 \int 0 e^{x^{2}} d x > 1 \int 0 d x \Rightarrow I > 1$

Option $(c)$ is correct.