To evaluate the integral from 0 to infinity of 26x*exp(x) dx, we start by recognizing that this is an improper integral since it has an infinite limit, specifically at the upper limit with the infinity symbol.
Step 1 - Setup:
Let us define the integral. We will perform integration by parts, which is going to be very useful with this formula:
Int(u dv) = u*v - Int(v du)
We have to choose our 'u' and 'dv'. We choose:
u = x; du = dx
dv = 26*exp(x) dx; v = 26*exp(x)
Step 2 - Integration by Parts:
Now we can perform the integration by parts, by evaluating u*v and subtracting the integral of v du:
= [u*v] - Int(v du) from 0 to ∞
= [x * 26*exp(x)] - ∫ [26*exp(x) * dx] from 0 to ∞
Step 3 - Compute Boundary Values and Second Integral:
Now we plug in the values of 0 and ∞ for x in [x * 26*exp(x)] and compute the remaining integral ∫ [26*exp(x) dx] from 0 to ∞.
At boundary x=0: 0 * 26 * exp(0) = 0
At boundary x=∞: ∞ * 26 * exp(∞) will go to ∞ since exp(∞) goes to ∞ faster than any polynomial function (so it doesn't converge).
Then the remaining integral ∫ [26*exp(x) dx] from 0 to ∞ equates to 26*exp(x) from 0 to ∞, which itself evaluates to ∞ since the exponential function grows without bounds.
So, the whole improper integral does not converge, and thus we can say its value is ∞.