Why is the output different for this integral in Mathematica?

  • Thread starter JulieK
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In summary, the integral \int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{b}dx is a complex expression where a and b are constants. By using the substitution ax = \sinh t, the integral can be simplified to \int \left(\frac{t}{\sinh t}\right)^b \frac{\cosh t}{a}\,dt. However, this still cannot be solved using standard mathematical functions and even the Wolfram online integrator cannot find a result. Mathematica also fails to provide a solution, giving a different output instead.
  • #1
JulieK
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What is this integral
[itex]\int\left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^{b}dx[/itex]
where a and b are constants.
 
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  • #2
The substitution [itex]ax = \sinh t[/itex] yields [tex]
\int \left(\frac{\mathrm{arcsinh}(ax)}{ax}\right)^b\,dx = \int \left(\frac{t}{\sinh t}\right)^b \frac{\cosh t}{a}\,dt \\
= \left[ \frac{1}{a(1-b)}\frac{t^b}{(\sinh t)^{b-1}}\right]
+ \frac{b}{a(b - 1)} \int \left(\frac{t}{\sinh t}\right)^{b-1}\,dt \\
[/tex] on integration by parts. Unfortunately that seems to be as far as one can get.
 
  • #3
The wonderful Wolfram online integrator can't do it, so there's not much hope...
 
  • #4
I confirm, Mathematica replies: "no result found in terms of standard mathematical functions" which is true in most cases.
 
  • #5
Just starting with Mathematica, I type in:
Code:
Integrate[((ArcSinh[a * x])/ a * x)^b, x]
and I get out:
Code:
\[Integral]((x ArcSinh[a x])/a)^b \[DifferentialD]x
Is there some reason I am getting a different output?
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve in a given interval. It is the inverse operation of differentiation and is used to calculate the total value of a function.

2. How is an integral different from a derivative?

An integral is the inverse operation of a derivative. While a derivative calculates the slope of a function at a specific point, an integral calculates the total value of the function over a given interval.

3. What is the purpose of using integrals?

Integrals are used in various fields of science, such as physics, engineering, and economics, to calculate the total value of a function. They are also used to solve problems involving area, volume, and motion.

4. How do you solve an integral?

To solve an integral, you need to use integration techniques such as substitution, integration by parts, or partial fractions. These techniques help to simplify the integral and make it easier to solve.

5. What is the difference between definite and indefinite integrals?

A definite integral has specific limits of integration, while an indefinite integral does not have any limits. In other words, a definite integral gives a specific value as the result, while an indefinite integral gives a function as the result.

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