Can Every Function Be Integrated?

Since the integral is defined by taking the area under the curve, an integral can be taken of any continuous function, because the area can be found.

However, it is not always possible to find the indefinite integral of a function by basic integration techniques. Some functions like $$\int e^{x^2} \, dx$$ cannot have an integral taken of them -- there actually isn't anyway to write the antiderivative of this function in terms of familiar common functions! However, using approximation by Riemann sums, a definite integral can be found of this function, or any function.