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I don't know how to evaluate this integral

$$\int_{0}^{1} \frac{\\3^x}{x^2}\mathrm dx $$

Could you give me advice on how I can do it because I am actually puzzled?

I tried to integrate by parts, but in finally, I had the integral

$$\int_{0}^{1} {e^x\ln x}dx $$

I didn't know how to evaluate it too.

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  • $\begingroup$ Sorry for my English in advance. $\endgroup$ – Artem Konovalov Nov 13 '18 at 18:58
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    $\begingroup$ The integral diverges. $\endgroup$ – projectilemotion Nov 13 '18 at 19:00
  • $\begingroup$ could you explain how you detected it? $\endgroup$ – Artem Konovalov Nov 14 '18 at 9:04
  • $\begingroup$ In the same way as WillM's answer. $\endgroup$ – projectilemotion Nov 14 '18 at 10:02
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Since $3^x \geq 1$ for $x$ between zero and $1$ and since $\int x^{-2} dx = -x^{-1},$ it follow that the integral is divergent.

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Sagecell.sagemath.org can do integrals for you. entering in the code:

var('y')

y = 3^x/x^2

integral(y, x, 0, 1)

returns the error:

ValueError: Integral is divergent.

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    $\begingroup$ Although this answer is technically correct, I th ink that this is a really bad way for a student to approach this problem. $\endgroup$ – user296602 Nov 13 '18 at 19:24
  • $\begingroup$ @T. Bongers: Im not suggesting that knowing the answer is sufficient. For someone struggling to formulate a plan of attack, it does help to know if they should be searching for a positive or negative result. $\endgroup$ – David Diaz Nov 13 '18 at 20:17
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    $\begingroup$ In that case, wouldn't this be more appropriate as a comment than an answer? $\endgroup$ – user296602 Nov 13 '18 at 20:19

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