>>11534770Answer: 5?/2
Solution:
From the problem:
cola = 10
burger = 5
beer = 1
hotdog = x
integral = I
So that we are solving
I = integral[(10sinx)dx/(2x)] from x = 0 to inf
? = 5*integral[(sinxdx)/x] from x = 0 to inf
Use the sine version of the Euler's Integral as shown below
?(s)*sin(s?)/[n(|p|)^s] = integral{[nx^(ns-1)][e^(-ax^n)][sin(bx^n)]dx} from x = 0 to inf
where p = a+bi, |p| = sqrt(a^2 + b^2), tan? = b/a
To solve this, set the following:
a = 0, b = 1, n = 1
|p| = sqrt(a^2 + b^2) = 1
tan? = b/a = 1/0 -> ? = ?/2
Thus, we get
?(s)*sin(s?/2) = integral[(sinxdx)/x^(s-1)] from x = 0 to inf
To evaluate LHS, we use the identity
?(s)?(1-s) = ?/sin(s?)
?hus,
{?/[sin(s?)*?(1-s)]}sin(s?/2) = integral[(sinxdx)/x^(s-1)] from x = 0 to inf
{s?/[sin(s?)*?(1-s)]}[sin(s?/2)/(s?/2)]*(?/2) = integral [(sinxdx)/x^(s-1)] from x = 0 to inf
Take the limit of both sides as s approaches 0, we get
?/2 = integral[(sinxdx)/x] from x = 0 to inf
Finally:
I = 5?/2
