antiderivative of 2sin(x^2)dx?

Question

this is also on the definite integral of pi/2 to pi........im so stuck........any help will be greatly appreciated.....

Answer 1

The antiderivative of sin (x²) cannot be written in terms of elementary functions. The only way to solve the problem would be to use numerical integration, or to represent the integrand using a Taylor series. Judging from the context of the problem though, I suspect that your problem is asking you for the integral of (sin x)², not sin (x²), which can be done as follows:

[π/2, π]∫2 sin² x dx
[π/2, π]∫1- cos (2x) dx (from the identity cos (2x)=cos² x - sin² x = 1-2 sin² x)
x - sin (2x)/2 | [π/2, π]
(π-sin (2π)/2) - (π/2 - sin π/2)
(π-0) - (π/2-0)
π/2

Answer 2

The anti-derivative of sin(x^2) cannot be expressed in terms of elementary functions. If you want to integrate 2sin(x^2)dx from pi/2 to pi, you must approximate it by some numerical method: Riemann sum, trapezoid rule, or Simpson's rule. They will give you a numerical approximation (calculators just do many terms to get it good enough to the number of digits they need to display).

If you meant 2x*sin(x^2) or 2*sin^2(x), then that'd be a different case. The first can be done by letting u = x^2, the latter can be done by using the trig identity: sin^2(x) = (1 - cos(2x))/2.

Answer 3

The answer to this problem is actually what they call a Fresnel Integral

for sine
S(x)==int(sin(pi*t^2/2 dt)
C(x)==int(sin(pi*t^2/2) dt)


S(x)dlx==cos(pi*x^2)/pi+x*S(x)


evaluation 2*sin(x^2)

2^(1/2)*pi^(1/2)*FresnelS(2^(1/2)/pi^(1/2)*x)

and the evaluation between pi/2 to pi is

FresnelS(2^(1/2)*pi^(1/2))*2^(1/2)*pi^(1/2)-FresnelS(1/2*2^(1/2)*pi^(1/2))*2^(1/2)*pi^(1/2)