One True or False question that was on my Calc II Final needs comnfirmation,,,?

Question

The statement was "The function f(x)=e^(x^2) has no antiderivative, True or False?

Please somebody help me with the answer. I said False, but my teacher said true.

Answer 1

It is true that this function has no explicit antiderivative that can be expressed by a finite number of functions.

(This is the case unless you are considering the "imaginary error function" as the answerer directly below me answered. However, I would be shocked if you are familiar with the erfi(x) function, so for you're purposes, the answer is still "true." Although the next answerer is absolutely correct and informative, I imagine that your professor was merely trying to ask if you could use some method, such as integration by parts, to integrate the function by hand, and the answer is "no.")

Answer 2

This function has an antiderivative, because any continuous function has an antiderivative.

However, the antiderivative of e^(x^2) cannot be written in terms of elementary functions. The antiderivative of e^(x^2) is just a completely different function than cannot be obtained using a finite combination the functions you usually use.

If you like, you can write the antiderivative of e^(x^2) as a power series. (This part will not make sense unless you studied power series in Calc II; some courses do and some do not.)

Recall that

e^x = Σ((x^n) / n!),

where the sum is taken from 0 to infinity. Plugging in x^2, we get

e^(x^2) = Σ((x^2n) / n!)

Integrating term-by-term gives

∫e^(x^2) = C + Σx^(2n + 1) / (n! * (2n + 1)),

where the sum is again taken from 0 to infinity.

This is a perfectly good function, as the power series converges everywhere; so this is one way to write the antiderivative of e^(x^2).

Answer 3

The function has an antiderivative. However, the antiderivative cannot be expressed in terms of a finite number of the functions you customarily study in calculus.

Every continuous function defined on a closed, bounded interval has an antiderivative-- its integral. That's what the Fundamental Theorem of Calculus says.

Answer 4

The function f(x) = e^(x²) is continuous everywhere,
so by the fundamental theorem of calculus,
it has an antiderivative. The problem is
that the antiderivative cannot be expressed in
terms of elementary functions. It defines a
higher function which is a constant multiple
of the imaginary error function.
I'm afraid your teacher was right.

Answer 5

The antiderivative of e^(x^2) cannot be expressed using a finite number of elementary functions, however one may use power series to arrive some sort of an answer.

Answer 6

That's not true.
The antiderivative is
sqrt(π) / 2 * erfi(x)

Write exp(x^2) as a power series
= ∑ x^(2n) / n!
integral = ∑ x^(2n+1) / [(2n+1)*n!]

Now consider the maclaurin series for
sqrt(π) / 2 * erf(x) = ∑ (-1)^n * x^(2n+1) / [(2n+1)*n!]

Replacing x by i*x
sqrt(π) / 2 * erf(ix) = ∑ (-1)^n * (ix)^(2n+1) / [(2n+1)*n!]
= ∑ (-1)^n * (i)^(2n+1) *x^(2n+1) / [(2n+1)*n!]
i^(2n)*i = (-1)^n * i

So sqrt(π) / 2 * erf(ix) = i * ∑ x^(2n+1) / [(2n+1)*n!]
or ∑ x^(2n+1) / [(2n+1)*n!] = sqrt(π) / (2i) * erf(ix)
= sqrt(π) / 2 * erfi(x)

It may appear that the second to last line is imaginary, but the i actually cancels off. It's a real antiderivative.

Answer 7

♣ yes, it has no anti-derivative!
You have just to know better about this function!
Nothing more!
There are several more functions with no anti-derivatives; and the teacher should have listed them in your classes!

♦ Guys do not mix integral with anti-derivative; the teacher is right!