Integrate sin(a*x^2) from -a to a

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  • #1
maze
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Hey all, this should be a simple integral, but I can't seem to solve it. I tried some simple u-subs and integration by parts, but to no avail. Also, maple throws out some Fresnel function.

[tex]\int_{-a}^{a} x^{2} cos\left( a x^{2} \right) dx[/tex]
 
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  • #2
Well, that should tell you it doesn't have an anti-derivative expressible in terms of elementary functions.
 
  • #3
I was hoping that perhaps it had a nice representation due to the limits of integration.

edit: I also don't particularly trust maple, as there have been times in the past where it couldn't do integrals that I could. (or perhaps I am using it wrong)
 
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  • #4
maze said:
I was hoping that perhaps it had a nice representation due to the limits of integration.
From the looks of it, it's symmetric, so you can change the limits of integration from 0 to a and multiplying your integral by 2. Idk if that helps at all or not :p
 
  • #5
This is not a homework question. I am not a student. Please move the question back to the appropriate forum.
 
  • #6
Well if it were an odd function, we would be able to conclude that the answer is 0 without having to work out the anti-derivative. But this is an even function.

By the way, if you want your thread to be moved back you should request a PF mentor by private messaging to do so.
 
  • #7
>> syms x;
>> f = x^2*cos(x^2);
>> int (f,x)

ans =

1/2*sin(x^2)*x-1/4*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2)*x)


>> int(f,x,-1,1)

ans =

sin(1)-1/2*2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2))

Matlab did same ><
I wanted to prove that Matlab is better

"edit: I also don't particularly trust maple, as there have been times in the past where it couldn't do integrals that I could. (or perhaps I am using it wrong)"
sometimes, you need to put the function in integrable form ... (computer's dun think)
 

1) What is the formula for integrating sin(a*x^2) from -a to a?

The formula for integrating sin(a*x^2) from -a to a is:

∫sin(a*x^2) dx = -1/2a * (cos(a*x^2) + 1) + C

Where C is the constant of integration.

2) How do you solve an integral with a trigonometric function like sin(a*x^2)?

To solve an integral with a trigonometric function like sin(a*x^2), you can use the substitution method. Let u = a*x^2 and then solve for dx in terms of du. The integral then becomes:

∫sin(u) * (1/2√a) * du

Which can be easily integrated using the power rule for integrals.

3) What is the significance of the limits of integration (-a to a) in this integral?

The limits of integration (-a to a) represent the range of values for which the integral is being evaluated. In this case, the limits of integration represent the area under the curve of sin(a*x^2) from -a to a on the x-axis.

4) Can this integral be evaluated using any other methods besides substitution?

Yes, this integral can also be evaluated using the symmetry property of sine, where ∫sin(x) dx from -a to a = 0. By substituting u = a*x^2, we can rewrite the integral as:

∫sin(u) * (1/2√a) * du

Which becomes 0 when evaluated from -a to a.

5) What are some real-world applications of this integral?

This integral has many applications in physics and engineering, particularly in the study of vibrations and waves. It can also be used in the analysis of electric and magnetic fields, as well as in probability and statistics. Additionally, it can be used to model the behavior of certain biological systems, such as the growth of cells or the spread of diseases.

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