Thou shalt generalize it first!

Calculus Level 5

$\large \displaystyle \int_{0}^{\infty}e^{-x^{2}}\cos^{3}{(x)}\mathbb{d}x =\dfrac{\pi^{\frac{1}{A}}}{B} \left[\eta e^{-\frac{C}{D}}+e^{-\frac{E}{F}}\right]$

If the above integral is true for positive integers $A,B,C,D,E,F,\eta$ , where $\gcd(C,D) = \gcd(E,F) = 1$ , find the value of $A+B+C+D+E+F-\eta$ .

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The answer is 25.

1 solution

Kunal Gupta
Sep 28, 2015

The answer is: $\huge \dfrac{\sqrt{\pi}}{8}\left[3 e^{-\frac{1}{4}}+e^{-\frac{9}{4}}\right]$ Ok, the solution is as follows!
Consider, $I(a) = \displaystyle \int_{0}^{\infty}e^{-x^{2}}\cos{(ax)}\mathbb{d}x$ Differentiating w.r.t $a$ , we get:
$I'(a) = \displaystyle -\int_{0}^{\infty}xe^{-x^{2}}\sin{(ax)}\mathbb{d}x$ Integrating by parts, we get
$I'(a) =-\dfrac{a}{2}I(a)$ Solving the ODE,we get: $\ln(I(a)) =-\dfrac{a^2}{4}+c$ Using the fact of the Gaussian Integral; $I(0) =\dfrac{\sqrt{\pi}}{2}$ We get: $I(a) =\dfrac{\sqrt{\pi}}{2}e^{-\frac{a^2}{4}}$ Also, $\cos^{3}(x) = \dfrac{3}{4}\cos(x) +\dfrac{1}{4}\cos(3x)$
The integral can be easily conjured up, to get the answer: $\huge \dfrac{\sqrt{\pi}}{8}\left[3 e^{-\frac{1}{4}}+e^{-\frac{9}{4}}\right]$ $\large \mathbb{Q.E.D}$

Can you show the relevant workings please? Thanks