I have the following problem: How do i integrate this: $ \int^1_0 e^{x^2+3 } de $ ? Whats putting me off is the de, i don't know how to integrate over the e-function...
According to Wolfram Alpha, it's the function itself: $ e^{x^2+3 } $
But how do i get there? I can't find a solution anywhere.
Think of it this way: it does not really matter what the variable you are integrating with respect to is denoted by. All following integrals represent the same thing - the same number: $$\int_0^1x^ndx=\int_0^1y^ndy=\int_0^1a^nda=\int_0^1e^nde=\int_0^1\otimes^nd\otimes$$
Now set $n=x^2+3$, this is just a constant with respect to $e$. Also to distract your mind from the conventional treatment of $e$ call it $t$ temporarely: $e=t$. Thus you arrive to $$\int_0^1t^ndt=\frac{1}{n+1}=\frac{1}{x^3+4}$$
This is an example of an irritating notation or perhaps an exercise to teach students that it is irrelevant how we denote variables.