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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

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    You are using linear interpolation between two not-so-close points, which will certainly result in a very rough approx. BTW, how did you obtain the four given approx ? – 2017-01-19
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    These 4 approximations were provided in the question. – 2017-01-19
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    Ok ... in this case, it may be the expected answer. But you should be interested in studying a method (or several ones ...) that lead to an approximation of any "reasonable" integral, with as high precision as you wish ... – 2017-01-19
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    Could you provide some information about the course, and perhaps also provide us with some methods (if any) given in that course. That could give us an idea of the intended method to use. – 2017-01-19

1 Answers 1

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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

$$0.229897$$