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

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.

• Sorry for my English in advance. – Artem Konovalov Nov 13 '18 at 18:58
• The integral diverges. – projectilemotion Nov 13 '18 at 19:00
• could you explain how you detected it? – Artem Konovalov Nov 14 '18 at 9:04
• In the same way as WillM's answer. – projectilemotion Nov 14 '18 at 10:02

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.

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.

• Although this answer is technically correct, I th ink that this is a really bad way for a student to approach this problem. – user296602 Nov 13 '18 at 19:24
• @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. – David Diaz Nov 13 '18 at 20:17
• In that case, wouldn't this be more appropriate as a comment than an answer? – user296602 Nov 13 '18 at 20:19