# Thread: Integral - infinite limit

1. ## Integral - infinite limit

$\displaystyle \displaystyle\int^\infty_1 \frac{1}{x^5}\,dx$

How is this calculated?

What is the name of these types of integrals (with infinite in limits and fractions in integrand)?

2. Originally Posted by anon_404
$\displaystyle \displaystyle\int^\infty_1 \frac{1}{x^5}\,dx$

How is this calculated?

What is the name of these types of integrals (with infinite in limits and fractions in integrand)?

This is known as an improper integral because the upper bound is infinity

$\displaystyle \int_a^\infty f(x)dx=\lim_{b\rightarrow\infty}\int_a^b f(x)dx$

$\displaystyle =\lim_{b\rightarrow\infty}F(b)-F(a)$

where $\displaystyle F'(x)=f(x)$

3. How is it calculated?

4. My previous post tells you to evaluate the antiderivative, plug in a variable for the uppper limit and then subtract the antiderivative evaluated at the lower limit, and take the limit as your variable goes to infinity

$\displaystyle \displaystyle\int \frac{1}{x^5}dx=-\frac{1}{4x^4}+c$

You should be able to solve your question now

5. Originally Posted by anon_404
How is it calculated?
The first step is to write $\displaystyle \lim_{\alpha \rightarrow + \infty} \int_1^{\alpha} \frac{1}{x^5} \, dx$.

6. Perhaps it would help you to write $\displaystyle \frac{1}{x^5}$ as $\displaystyle x^{-5}$. Do you know the anti-derivative of $\displaystyle x^n$?