Approximate $I(x)=\int_{0}^{x}(\cos t)e^\frac{-t^2}{10}\,\mathrm dt$ for $x=\frac{5\pi}4$, given $I(x)$ for $x=\frac\pi2,\pi,\frac{3\pi}2,2\pi$

Question

Approximate value of the integral $I(x)=\int_{0}^{x}(\cos t)e^\frac{-t^2}{10}\,\mathrm dt$ are given below

$I(\frac{\pi}{2}) = 0.95$,

$I({\pi}) = 0.44$,

$I(\frac{3\pi}{2}) = 0.18$,

$I({2\pi}) = 0.22$

Then evaluate approximate value of the integral

$$\int_{0}^{\frac{5\pi}{4}}(\cos t)e^\frac{-t^2}{10}\,\mathrm dt$$

I could see that value of $\frac{5\pi}{4}$ can be written as $\pi + \frac{\pi}{4}$. hence the value of above integral should lie between 0.44 and 0.18

and also I could see that $(\pi + \frac{3\pi}{2} )/2 = \frac{5\pi}{4}$

Hence I approximate the value as $\frac{(0.44+0.18)}{2}$ which evaluates to 0.31, but this is neither the correct answer nor seems to be a correct approach. Please provide any hints or properties of definite integration, by which I could evaluate the above integral

Answer 1

Just for the sake of curiosity, let me tell you that your integral is actually a well known integral in terms of the so called Error Function.

You may not know it, but, as I said, this is just for the sake of curiosity.

Indeed your integral is nothing but

$$\int \cos(t)\ e^{-at^2}\ \text{d}t$$

Where $a = 1/10$.

The integral can be solved and it gives

$$\int \cos(t)\ e^{-at^2}\ \text{d}t = \frac{\sqrt{\pi } e^{-\frac{1}{4 a}} \left(\text{erf}\left(\frac{2 a t-i}{2 \sqrt{a}}\right)+i \text{erfi}\left(\frac{1-2 i a t}{2 \sqrt{a}}\right)\right)}{4 \sqrt{a}}$$

More particularly:

$$\int_0^M \cos(t)\ e^{-at^2}\ \text{d}t = \frac{\sqrt{\pi } e^{-\frac{1}{4 a}} \left(\text{erf}\left(\frac{2 a M-i}{2 \sqrt{a}}\right)+i \text{erfi}\left(\frac{1-2 i a M}{2 \sqrt{a}}\right)\right)}{4 \sqrt{a}}$$

And even more particularly

$$\int_0^M \cos(t) e^{-t^2/10}\ \text{d}t = \frac{\sqrt{\frac{5 \pi }{2}} \left(\text{erf}\left(\frac{M+5 i}{\sqrt{10}}\right)-i \text{erfi}\left(\frac{5+i M}{\sqrt{10}}\right)\right)}{2 e^{5/2}}$$

In your case $M = \frac{5\pi}{4}$, that is

$$\frac{\sqrt{\frac{5 \pi }{2}} \left(\text{erf}\left(\frac{M+5 i}{\sqrt{10}}\right)-i \text{erfi}\left(\frac{5+i M}{\sqrt{10}}\right)\right)}{2 e^{5/2}}\bigg|_{M = \frac{5\pi}{4}} = \frac{\sqrt{\frac{5 \pi }{2}} \left(\text{erf}\left(\frac{1}{4} \sqrt{\frac{5}{2}} (\pi +4 i)\right)-i \text{erfi}\left(\frac{5+\frac{5 i \pi }{4}}{\sqrt{10}}\right)\right)}{2 e^{5/2}}$$

With a numerical value equal to

$$0.229897$$