Integrate the Taylor series for ex term by term, and find the value of the constant for

Question:

Integrate the Taylor series for ex term by term, and find the value of the constant for which the integral matches the function ex.
When a function has a well-behaved Taylor series, we can find a Taylor series for some integrals by integrating term by term (proving that these integrals converge to the correct answer requires methods from more advanced courses).