Another infinite integral!
Published in
Jul 27, 2023
Before looking at the solution, give the problem a go!
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Similar to our original infinite integral, we want to try to write this in a more compact way. Let us see what this looks like in terms of exponents:
Now, what happens if we take an x on the outside? We get the following:
In fact, let us keep taking these out. We will see a pattern emerge in just a second:
If we keep going, we can see that we will clearly multiply by the next integer on the denominator then add it on. Therefore, we get that our integral is equal to the following:
However, let us remind ourselves of the Taylor series for e^x:
Our exponent is the same evaluated at x=1 except we are starting from 1, hence our integral is nothing but:
And therefore:
