how do you integrate sin (x^2) dx?
There is a solution given here that uses complex numbers but gives a simple real answer which is correct:
http://answers.yahoo.com/question/index?qid=20070116202856AA7DM5R
You might be able to get the same answer using integration by parts
The answer is
(x/2) - (1/4)sin(2x)
I don't see any easier way to get it than the "Euler" proof in the link I gave you
thank you for your answer, but this example problem is not the same as mine. the example problem is (sin (x))^2, in other words the whole quantity is squared. mine is just sin of x^2. any suggestions here?
To integrate sin(x^2) dx, there is no known elementary function to represent its antiderivative. However, you can approximate the integral using numerical methods or specialized functions.
One numerical method to approximate the integral of a function is the Simpson's Rule. This method involves dividing the interval of integration into smaller subintervals and approximating the integral as a weighted sum of function values at specific points within each subinterval.
Here's how you can use Simpson's Rule to approximate the integral of sin(x^2) dx:
1. Choose the interval of integration, from a to b, where you want to find the integral.
2. Decide on the number of subdivisions, n, for Simpson's Rule. The larger the value of n, the more accurate the approximation.
3. Determine the width of each subdivision, h, by calculating (b - a) / n.
4. Define the function f(x) = sin(x^2).
5. Initialize the variables sum1, sum2, and sum3 to 0.
6. Iterate from i = 1 to n - 1, calculating the following:
- For the first term in the sum, if i is odd: sum1 += f(a + i * h)
- For the second term in the sum: sum2 += f(a + i * h)
- For the third term in the sum, if i is even: sum3 += f(a + i * h)
7. Calculate the approximation of the integral using Simpson's Rule formula:
- integral ≈ (h / 3) * (f(a) + f(b) + 4 * sum1 + 2 * sum2 + 4 * sum3)
This approximation method yields an estimated value for the integral. Keep in mind that the accuracy depends on the chosen value of n. By increasing n, the approximation will become closer to the exact value, but at the cost of more computations.
Alternatively, you can use numerical software or functions, such as the SciPy library in Python or specialized mathematical software, to evaluate the integral of sin(x^2) dx more accurately.