`int_1^oo 6/x^4 dx` Determine whether the integral diverges or converges. Evaluate the integral if it converges.

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An integral in which one of the limits of integration is infinity is an improper integral.  Because we cannot find the definite integral using infinity (since it is not an actual value), we will need to rewrite the improper integral as a limit, shown below:

`int_1^oo 6/(x^4)dx=lim_(n->oo)int_1^n 6/(x^4)dx`

We can...

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An integral in which one of the limits of integration is infinity is an improper integral.  Because we cannot find the definite integral using infinity (since it is not an actual value), we will need to rewrite the improper integral as a limit, shown below:

`int_1^oo 6/(x^4)dx=lim_(n->oo)int_1^n 6/(x^4)dx`

We can now take the integral first and the limit second.  We can find the integral using the power rule:

`lim_(n->oo)int_1^n 6/(x^4)dx=lim_(n->oo)int_1^n 6x^-4dx=lim_(n->oo)[6*x^-3/-3]_1^n=lim_(n->oo)[-2/x^3]_1^n`

Now we simply need to evaluate the limit. As n approaches infinity, the fraction will approach 0:

`lim_(n->oo)-2/n^3-(-2)/1^3=lim_(n->oo)-2/n^3+2=0+2=2`

Thus the integral converges to a value of 2.

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