Computing Improper Integrals

Here are a number of examples illustrating how Mathematica may be used to determine the value of an improper integral, or simply to determine if an improper integral converges or diverges.

(∫_1^2)(1/(Sqrt[(x^2)-1]))\[DifferentialD]x

(∫_5^∞)(1/(x Log[x] (Log[Log[x]])))\[DifferentialD]x

(∫_5^∞)(1/(x Log[x] (((Log[Log[x]]))^1.1)))\[DifferentialD]x

(∫_(-∞)^∞)(x^4)*Exp[-(x^2)]\[DifferentialD]x

(∫_0^∞)(1/(1+x+(x^2)+(x^3)))\[DifferentialD]x

At times we may wish to deal with symbolic constants and find that the result of a particular integration depends upon  the constants assuming values within a particular interval.  For example, a constant is assumed to be positive.  In such cases we may use the Assumption condition.

Integrate[Exp[-a*x], {x,0,∞}, Assumptions-> a>0 ]

Integrate[(1/(Sqrt[b*x+(x^2)])), {x,0,1}, Assumptions-> b>0 ]

Let's also note that Mathematica accepts the factorial function:

n=10

Plot[((n!)/(Exp[n])), {n,3, 11}]