Question

How do I find the antiderivative of `e^(2x) + 1`?

Answer 1

I would use the idea of integral (indefinite) and the techniques connected with this procedure:
1) I can write:
`inte^(2x)+1dx`
2) I can use the fact the the integral of a sum is equal to the sum of the integrals, giving:
`inte^(2x)dx+int1dx`
3) I can use the fact that the integral of the exponential is equal to itself (but here we have to consider the exponent `2x` as well) and that `1` can be written as `x^0`;
`inte^(2x)dx+intx^0dx=`
`e^(2x)/2+x+c`

I also evaluate the integral of `x^0` using the fact that the integral of `x^n` is `x^(n+1)/(n+1)`

4) You can now check the result (the anti-derivative) obtained above deriving it to see if it gives the initial function `e^(2x)+1`.