How to find limit of $\lim_{N \to \infty} \int_{N}^{e^N} xe^{-x^{2017}}dx$

Question

$$\lim_{N \to \infty} \int_{N}^{e^N} xe^{-x^{2017}}dx$$

I know limits for improper integrals of the form: $$\int_{a}^b f(x)dx$$ where $b \to \infty$.

But this appears to be of the form: $$\int_{\infty}^\infty f(x)dx$$

Answer 1

Hint. Note that for $N>1$, $$0<\int_{N}^{e^N} xe^{-x^{2017}}dx< \int_{N}^{+\infty} xe^{-x^{2017}}dx\leq \int_{N}^{+\infty} xe^{-x}dx=(N+1)e^{-N}$$ then use the Squeeze Theorem.

Answer 2

HINT:

$$\left|\int_N^{e^N}xe^{-x^{2017}}\,dx\right|\le (e^N-N)e^Ne^{-N^{2017}}$$