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Evaluate
[0⁰:
assuming that the integral converges.
26xe* dx

User Ssloan
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1 Answer

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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 ∞.

User Garzj
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