First slide

Evaluation of definite integrals

Question

The value of $lim_{x \to \infty} \frac{{(\int_{0}^{x} e^{x} d x)}^{2}}{\int_{0}^{x} e^{2 x^{2}} d x}$ is

Moderate

A

1
B

2
C

3
D

0

Solution

$\begin{array}{l} lim_{x \to \infty} \frac{{(\int_{0}^{x} e^{x} d x)}^{2}}{\int_{0}^{x} e^{2} x^{2} d x} = lim_{x \to \infty} \frac{{(e^{x} - 1)}^{2}}{\int_{0}^{x} e^{2 x^{2}} d x} \\ = lim_{x \to \infty} \frac{2 (e^{x} - 1) e^{x}}{e^{2 x^{2}}} = 2 lim_{x \to \infty} \frac{1 - e^{- x}}{e^{2 x^{2} - 2 x}} = 0 \end{array}$

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