Question

${\int }_1^5{e}^{(}{x}^2)dx$ lies in which of the following interval ?

• A. $(4e,4e^{25})$
• B. $(0,4e^{25})$
• C. $BothA\&B$
• D. $Ca{n}^{\prime }tbedetermined.$

Answer 1

The correct option is C $BothA\&B$

Well, the given question is a tricky one. But if you know the required concepts and know how to apply them it’s quite easy. We know that the given integrand is non integrable. If you don’t please go through your indefinite integrations notes once.
So the given function can’t be integrated. But still we can find the interval in which the given integral lies.
We have seen that if there is a monotonically increasing function f(x) then we can say -
$(b-a)f\left(a\right)\le {\int }_a^{b}f\left(x\right)dx\le (b-a)f\left(b\right);whereb>a$
We’ll apply the same concept if the given integrand is monotonically increasing.
Remember how to check whether the function is monotonically increasing or not ?
Yes! We’ll differentiate and see whether the differentiation of it is greater than equal to zero or not.
$Letf\left(x\right)=e^{x^2}{f}^{\prime }\left(x\right)=e^{x^2}.2x$
Is it greater than zero ?
Not always, right?
Oops! It’s not a monotonically increasing function.
But wait ! Why does it have to be always monotonically increasing? We just have to consider the part of it which comes under the given limits i.e. $xϵ[1,5]$
& we know that for all positive values of x f’(x) is positive.
$f^{\prime }\left(x\right)=e^{(}{x}^2).2x>0\forall x>0So,f^{\prime }(x)>0;xϵ[1,5]$
From this we can say that the given function is monotonically increasing in the given limits and the following expression will hold true.
$(b-a)f\left(a\right)\le {\int }_a^{b}f\left(x\right)dx\le (b-a)f\left(b\right);whereb>aTherefore,(5-1)f\left(1\right)\le {\int }_1^5{e}^{x^2}dx\le (5-1)f\left(5\right))I.e,4.e\le {\int }_1^5{e}^{x^2}dx\le 4.e^{25}$
So the given integral lies in the interval $[-4.e^{25}]$
We can see that the other given options contains this interval also. So, the value of the integral must lie in both the intervals.