# Integral of sin(x^2)

by -EquinoX-
Tags: integral, sinx2
 P: 577 1. The problem statement, all variables and given/known data what is the integral of sin(x^2) dx? 2. Relevant equations 3. The attempt at a solution
 Sci Advisor HW Helper Thanks P: 25,243 It's one of those integrals like e^(-x^2) that doesn't have an elementary antiderivative. Why are you asking?
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,357 That does not have an integral in terms of elementary functions.
 P: 577 Integral of sin(x^2) as I am asked to calculate integral from y^2 to 25 of y * sin(x^2) dx and I am stuck with the sin(x^2), the y can be treated as a constant. Can you give some help? tried to use some online help here and the result was just bizarre: http://www.numberempire.com/integralcalculator.php
 Sci Advisor HW Helper Thanks P: 25,243 Is that REALLY the whole problem? Or is there more you aren't telling us about?
 P: 577 this is the whole problem: integral from 0 to 5, integral from y^2 to 25 of y * sin(x^2) dx dy It's a double integral
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,357 So, since you cannot integrate sin(x2) in elementary functions, reverse the order of integration, as I suggested.
 P: 577 ok so after reversing it I have integral from 0 to 25 , integral from 0 to sqrt(x) of y sin(x^2) dy dx. Doing the first integration results in integral from 0 to 25 of (sin(x^2)*x)/2 and I got -cos(x^2)/4 evaluated from 0 to 25. Is this correct so far?
P: 2
 Quote by -EquinoX- 1. The problem statement, all variables and given/known data what is the integral of sin(x^2) dx? 2. Relevant equations 3. The attempt at a solution
how i can evaluate it
gi me now
 P: 2 please give me a solution no to me many thanks to you.
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,357 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, $sin(x^2)$ does NOT have an elementary anti-derivative. After EquinoX told us that the problem was really $$\int_{y= 0}^5\int_{x= y^2}^{25} y sin(x^2)dx dy$$ it was suggested that he reverse the order of integration. Doing that it becomes $$\int_{x= 0}^{25}}\int_{y= 0}^{\sqrt{x}} y sin(x^2)dy dx$$ $$= \frac{1}{2}\int_{x= 0}^{25}\left[y^2\right]_{y=0}^{\sqrt{x}} sin(x^2) dx$$ $$= \frac{1}{2}\int_{x= 0}^{25} x sin(x^2) dx$$ which can be integrated by using the substitution $u= x^2$: If $u= x^2$, du= 2x dx so x dx= (1/2)du. When x= 0, u= 0 and when x= 25, u= 625 so the integral is $$\frac{1}{4}\int_0^{625} sin(u) du= -\frac{1}{4}\left[cos(u)\right]_0^{625}$$ $$= -\frac{1}{4}(-0.984387)= 0.246097$$
 P: 607 According to Maple, it is a Fresnel S integral... $\int \sin(x^2)\,dx = \frac{\sqrt {2}\sqrt {\pi }}{2}\,{\rm S} \left( {\frac {\sqrt {2}x}{ \sqrt {\pi }}} \right)$
 P: 1 sin x^2 = 1 - cos 2x and we can use 1 and cos 2x seperatly and solve this problem.
Math
Emeritus
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,357 So you resurrected this thread from over a year ago just to say you did not understand it? The original question was to integrate $sin(x^2)$, NOT $sin^2(x)$ for which your solution would be appropriate. That was said back in November of 2009.