Differentiation under the integral sign problem

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Homework Statement


[itex]\int_1^2 \frac {e^x}{x}\,dx[/itex] Through the use of differentiation under the integral sign.

2. The attempt at a solution
Tried inputting a several times, each one resulting in another function without an elementary anti-derivative (example given below)
[itex]I(a)=\int_1^2 \frac {e^{-ax}}{x}\,dx[/itex]
[itex]I'(a)=\int_1^2 -e^{-ax}\,dx[/itex]
[itex]I'(a)=\frac {e^{-2a}}{a}-\frac {e^{-a}}{a}[/itex]
[itex]I(a)=\int \frac {e^{-2a}}{a}\,da-\int \frac {e^{-a}}{a}\,da[/itex]
Resulting in a non-elementary anti-derivative...

Other examples I've tried include:
  • [itex]I(a)=\int_1^2 \frac {e^x*sin(ax)}{x}[/itex]
  • Subsituting x for ln(b), resulting in [itex]\int_1^2 \frac {1}{b*ln(b)}\,db[/itex], then saying [itex]I(a)=\int_1^2 \frac {1-b^a}{b*ln(b)}\,db[/itex]
Both of these last two also result in a non-elementary anti-derivative
 

Answers and Replies

  • #2
member 587159

Homework Statement


[itex]\int_1^2 \frac {e^x}{x}\,dx[/itex] Through the use of differentiation under the integral sign.

2. The attempt at a solution
Tried inputting a several times, each one resulting in another function without an elementary anti-derivative (example given below)
[itex]I(a)=\int_1^2 \frac {e^{-ax}}{x}\,dx[/itex]
[itex]I'(a)=\int_1^2 -e^{-ax}\,dx[/itex]
[itex]I'(a)=\frac {e^{-2a}}{a}-\frac {e^{-a}}{a}[/itex]
[itex]I(a)=\int \frac {e^{-2a}}{a}\,da-\int \frac {e^{-a}}{a}\,da[/itex]
Resulting in a non-elementary anti-derivative...

Other examples I've tried include:
  • [itex]I(a)=\int_1^2 \frac {e^x*sin(ax)}{x}[/itex]
  • Subsituting x for ln(b), resulting in [itex]\int_1^2 \frac {1}{b*ln(b)}\,db[/itex], then saying [itex]I(a)=\int_1^2 \frac {1-b^a}{b*ln(b)}\,db[/itex]
Both of these last two also result in a non-elementary anti-derivative
Not every integral of a function can be written as elementary functions. Try to use numerical integration methods.
 
  • #3
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Not every integral of a function can be written as elementary functions. Try to use numerical integration methods.
If I understand you correctly, you're saying to use a different method to find the value of [itex]\int_1^2 \frac {e^x}{x}\,dx[/itex], rather than differentiation under the integral sign.

The question I was asked stated to use differentiation under the integral sign.

If I understand you wrong, could you rephrase your statement?
 
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  • #4
member 587159
I mean, you can't solve this integral by finding a primitive. You either use a numerical integration method (Simpson's method, for instance) or you express e^x/x as a power series. In that case, you can determine the power series of the primitive function and approximate the solution.
 
  • #5
SteamKing
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I mean, you can't solve this integral by finding a primitive. You either use a numerical integration method (Simpson's method, for instance) or you express e^x/x as a power series. In that case, you can determine the power series of the primitive function and approximate the solution.
You can use the Leibniz Rule to evaluate definite integrals without resorting to purely numerical approximations or using infinite series. That's what the point of the question is all about in the OP: how to use the Leibniz Rule in this fashion.
 
  • #6
Ray Vickson
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You can use the Leibniz Rule to evaluate definite integrals without resorting to purely numerical approximations or using infinite series. That's what the point of the question is all about in the OP: how to use the Leibniz Rule in this fashion.
It is clear that the OP knows how to use Leibniz' rule, and has made several attempts to apply it, but none of them seem to work.
 

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