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Chapter 1: Problem 6

Assuming that \(\int\left(\mathrm{e}^{\mathrm{x}} / \mathrm{x}\right) \mathrm{d} \mathrm{x}\) is not elementary (a theorem of Liouville), prove that \(\int 1 / \ln \mathrm{x} \mathrm{dx}\) is not elementary.

Short Answer

Expert verified
Question: Prove that the integral of \(\int \frac{1}{\ln x} dx\) is not elementary, given that the integral of \(\int \frac{e^x}{x} dx\) is not elementary (a theorem of Liouville). Answer: By applying the substitution method, we find that the transformed integral \(\int \frac{1}{\ln x} dx\) is equal to the given integral \(\int \frac{e^x}{x} dx\), which is not elementary according to Liouville's theorem. Therefore, it can be concluded that the integral \(\int \frac{1}{\ln x} dx\) is also not elementary.

Step by step solution

01

Substitution

We apply substitution method to simplify \(\int \frac{1}{\ln x} dx\). Let \(u = \ln x\). So, we have \(x = \mathrm{e}^u\). Differentiating both sides with respect to \(x\), we get: \(du = \frac{1}{x} dx\) or \(dx = x \, du\), as \(x = \mathrm{e}^u\) we get \(dx = \mathrm{e}^u \, du\). Now substitute these values into the original integral. The integral becomes: \(\int \frac{1}{\ln x} dx = \int \frac{1}{u} \cdot \mathrm{e}^u \, du\)
02

Comparing integral expressions

Now, we compare the new integral expression with the given integral that is not elementary: \(\int \frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{x}}\, \mathrm{d} \mathrm{x}\). Notice that the transformed integral has the same structure as the one given: \(\int \frac{1}{u} \cdot \mathrm{e}^u \, du = \int \frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{x}}\, \mathrm{d} \mathrm{x}\).
03

Conclusion

Since the transformed integral is equal to the given integral, and we know that the integral \(\int \frac{\mathrm{e}^{\mathrm{x}}}{\mathrm{x}}\, \mathrm{d} \mathrm{x}\) is not elementary (by Liouville's theorem), it implies that the integral \(\int \frac{1}{\ln x} dx\) is not elementary as well.

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Most popular questions from this chapter

Evaluate the following integrals : (i) \(\int \frac{1}{x \sqrt{\left(1+x^{6}\right)}} d x\) (ii) \(\int \frac{x d x}{(p x+q)^{3 / 2}}\) (iii) \(\int \frac{x^{5}}{\sqrt{\left(1+x^{2}\right.}} d x\) (iv) \(\int \frac{d x}{x^{3} \sqrt{x^{2}+1}}\)

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