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Problem on Integral Inequality | ISI – MSQMS – B, 2015

Try this problem from ISI MSQMS 2015 which involves the concept of Integral Inequality and real analysis. You can use the sequential hints provided to solve the problem.

Try this problem from ISI-MSQMS 2015 which involves the concept of Integral Inequality.

INTEGRAL INEQUALITY | ISI 2015 | MSQMS | PART B | PROBLEM 7b


Show that $1<\int_{0}^{1} e^{x^{2}} d x<e$

Key Concepts


Real Analysis

Inequality

Numbers

Check The Answer


But Try the Problem First…

ISI – MSQMS – B, 2015, Problem 7b

“INEQUALITIES: AN APPROACH THROUGH PROBLEMS BY BJ VENKATACHALA”

Try with Hints


We have to show that ,

$1<\int_{0}^{1} e^{x^{2}} d x<e$

$ 0< x <1$

It implies, $0 < x^2 <1$

Now with this reduced form of the equation why don’t you give it a try yourself, I am sure you can do it.

Thus, $ e^0 < e^{x^2} <e^1 $

i.e $1 < e^{x^2} <e $

So you are just one step away from solving your problem, go on………….

Therefore, Integrating the inequality with limits $0$ to $1$ we get, $\int\limits_0^1 \mathrm dx < \int\limits_0^1 e^{x^2} \mathrm dx < \int\limits_0^1e \mathrm dx$

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