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I came across a list of functions in my calculus textbook whose indefinite integral cannot be found. It was written that the integral $$\int \frac{\sin(x)}{x} dx$$ cannot be evaluated without any explanation as to why.

I did some research over the internet and found out that the definite integral $$\int_{0}^{\infty} \frac{\sin(x)}{x} dx$$ can be evaluated using Laplace transformation and is equal to $\pi /2$. But I still couldn't find answer to my original question. I read somewhere that the integral cannot be expressed using 'elementary functions'. A little help is appreciated, I'm in Calc 1 going advanced than my course but I am sorry if my post shows lack of research. Thank you!

John Doe
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QFTheorist
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    As we'd accept arbitrarily nested elementary functions, such a negative result is far from trivial or far from calc 1 level - you may compare it to showing that $\pi$ is transcendental when you just know how to solve quadratics ... – Hagen von Eitzen Apr 29 '18 at 15:06
  • https://math.stackexchange.com/questions/5248/evaluating-the-integral-int-0-infty-frac-sin-x-x-dx-frac-pi-2 – Karl Apr 29 '18 at 15:13
  • It depends on definition of elementary functions. One can however consider this integral as a new elementary function. – user Apr 29 '18 at 15:14
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    http://math.stanford.edu/~conrad/papers/elemint.pdf might help you (some of the more technical passages can be skipped if they're too advanced for you). The general result is called Liouville's theorem, or Liouville's impossibility theorem, or Liouville's theorem in differential algebra. – Chappers Apr 29 '18 at 15:27

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Since every continuous real function $ f(x) $ has its indefinite integral $ F(x) $ on its domain by Newton-Leibniz formula $$ F(x)=\int_a^xf(x)dx, \quad x\in [a, b] $$

But we cannot find an expression of $ F(x) $ using elementary functions and their composition of a finite number of arithmetic operations $(+ – × ÷)$, exponentials, logarithms, constants, and solutions of algebraic equations. Whereas, it doesn't mean that we cannot calculate them, by some certain methods such as using complex analysis, we can calculate the exact value of its definite integral on some proper interval.

Bach
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