Do I need to use McLaurin to solve the integral $ \int \frac{e^{3 x}}{x^3} dx$.

Question

I need to compute this primitive :

$$ \int \frac{e^{3 x}}{x^3} dx$$

but I don't know how to proceed. I tried an integration by part and get

$$ \int \frac{e^{3x}}{x} dx$$

but then I am stuck... Is there no way else than making a McLaurin development ?

Thanks

Answer 1

Always give a try to "the integrator"

https://www.wolframalpha.com/input/?i=integrate+exp(3*x)%2F(x%5E3)

A special function is needed for the analytic expression, see https://en.wikipedia.org/wiki/Exponential_integral

Answer 2

$$\int \frac{e^x}{x} dx$$ is not expressible in terms of elementary functions. Mathematicians define $$ Ei(x) = -\int_{-x}^{\infty} \frac{e^{-t}}{t} dt $$

Then one has that

$$\int_{- \infty }^{x} \frac{e^x}{x^3} dx $$ $$ = - \frac{1}{2}( x^{-2} + 3 x^{-1} ) e^{3x} -27Ei(3x) $$

(using Integration By Parts).