integral of sin(x^2) dx

i actually got the ans, but aftr differentiating the answer i am not getting back the question

let u=x^2

du= 2x dx

dx=du/2x
dx=du/2sqrt(u)

-1/4 cos u is the integral

http://www.numberempire.com/integralcalculator.php

This involves the error function, not an elementary problem. Hannah has made an error.

I'm sorry about that. Can you explain why there is an error Mr. Pursley? I genuinely don't understand how there is an error.

Your substitutions are good, but you dropped something. You should have watched the details:

∫sin(x^2) dx = ∫sin(u) / 2√u du

then how do we proceed ??

is their any solution to this problem?

To find the integral of sin(x^2) dx, you can try using a technique called integration by parts. However, this method can get quite complicated, and it might not yield a simple or easily recognizable antiderivative.

A more common approach to compute this integral is to use numerical methods, such as numerical integration or approximation algorithms. These methods employ techniques like Simpson's rule, the trapezoidal rule, or Monte Carlo integration. These techniques involve breaking down the integral into smaller intervals and calculating the function's value within each interval to approximate the integral.

Alternatively, if you're interested in evaluating this integral symbolically, you can use specialized mathematical software or online resources like Wolfram Alpha. Such tools employ advanced algorithms and techniques to handle more complex integrals, including those involving special functions.

As for your comment about differentiating the answer and not arriving at the original question, it is important to note that taking the derivative of an antiderivative will not always give you the original function back. Although the integral gives you one possible antiderivative, it is not always unique, and adding a constant of integration can yield an infinite number of other valid antiderivatives.

So, if you have already computed the integral and obtained an expression, there is no guarantee that taking the derivative of that expression will yield the original integrand. However, you can verify that the derivative of the integral you obtained matches the original integrand by confirming that they have the same values for specific input values.