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-EquinoX-
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Homework Statement
what is the integral of sin(x^2) dx?
-EquinoX- said:Homework Statement
what is the integral of sin(x^2) dx?
Homework Equations
The Attempt at a Solution
No, "sin x^2" MEANS sin(x^2) and cannot be integrated in that way. If your function is really (sin(x))^2= sin^2(x), you should have told us that immediately.anushyan88 said:sin x^2 = 1 - cos 2x
and we can use 1 and cos 2x seperatly and solve this problem.
Mstf_akkoc said:dx=cos(x^2)dm
micromass said:Why is this true?
Mstf_akkoc said:Actualy is not true. I have done a misteke when calculete the [tan(x^2)]'
2xdx=cos(x^2)dm is true
if I find a solution with this I ll write.
HallsofIvy said:Why have you suddenly jumped into this thread from back in February? Did you not read the previous posts? As said in the very first response, [itex]sin(x^2)[/itex] does NOT have an elementary anti-derivative.
After EquinoX told us that the problem was really
[tex]\int_{y= 0}^5\int_{x= y^2}^{25} y sin(x^2)dx dy[/tex]
it was suggested that he reverse the order of integration. Doing that it becomes
[tex]\int_{x= 0}^{25}\int_{y= 0}^{\sqrt{x}} y sin(x^2)dy dx[/tex]
[tex]= \frac{1}{2}\int_{x= 0}^{25}\left[y^2\right]_{y=0}^{\sqrt{x}} sin(x^2) dx[/tex]
[tex]= \frac{1}{2}\int_{x= 0}^{25} x sin(x^2) dx[/tex]
which can be integrated by using the substitution [itex]u= x^2[/itex]:
If [itex]u= x^2[/itex], du= 2x dx so x dx= (1/2)du. When x= 0, u= 0 and when x= 25, u= 625 so the integral is
[tex]\frac{1}{4}\int_0^{625} sin(u) du= -\frac{1}{4}\left[cos(u)\right]_0^{625}[/tex]
[tex]= -\frac{1}{4}(-0.984387)= 0.246097[/tex]
The integral of sin(x^2) dx cannot be expressed in terms of elementary functions. It is an unsolved integral known as the Fresnel integral.
Yes, there are various methods for approximating this integral, such as Simpson's rule or the Riemann sum.
Yes, numerical methods such as the trapezoidal rule or Gaussian quadrature can be used to approximate the value of this integral.
The integral of sin(x^2) dx is a special case of the Fresnel integral, which has applications in optics, diffraction, and quantum mechanics.
There are various mathematical software packages, such as Mathematica or MATLAB, that have built-in functions for evaluating this integral numerically or symbolically.