0 to 1/2∫(sin(x^2))dx
Recall that
sin^2 x = 1/2 - (1/2)cos (2x)
so...
∫(sin(x^2))dx from 0 to 1/2
= ∫(1/2 - (1/2)cos (2x) dx from 0 to 1/2
= x/2 - (1/4)sin (2x) from 0 to 1/2
= ......
you cannot integrate sin(x^2) using elementary functions.
Some numeric method is needed, or one of the special functions defined as integrals. In this case, the Fresnel Integral.
To evaluate the integral of sin(x^2), you can use a technique called numerical integration. Numerical integration involves approximating the integral using numerical methods, usually through the use of computer algorithms.
One common numerical integration technique is known as the trapezoidal rule. This method approximates the area under the curve by dividing the interval of integration into small trapezoids and summing up their areas. The more trapezoids you use, the more accurate the approximation becomes.
To apply the trapezoidal rule, follow these steps:
1. Determine the interval of integration. In this case, it is from 0 to 1/2.
2. Choose the number of trapezoids (n) to use in the approximation. The larger the value of n, the more accurate the result will be. Let's start with a relatively small value such as n = 10.
3. Divide the interval of integration into n equal subintervals. In this case, divide the interval from 0 to 1/2 into 10 equal subintervals. The width of each subinterval (Δx) would be (1/2 - 0)/10 = 1/20.
4. Evaluate the function sin(x^2) at each boundary point and each mid-point of the subintervals. This will give you the y-values that represent the heights of the trapezoids.
5. Apply the trapezoidal rule formula to each pair of adjacent points. The formula is (y1 + y2)/2 * Δx, where y1 and y2 are the heights of the adjacent points, and Δx is the width of the subinterval.
6. Sum up all the areas of the trapezoids to get the approximate value of the integral.
Using these steps, you can now perform the numerical integration to approximate the value of the integral of sin(x^2) from 0 to 1/2.