This sigma notation has a form of riemann sums; hence, we can convert this sigma notation into an integral. From this sigma notation, we can get some key informations.
i) a = pi/2
ii) b = pi/2 + pi/2 = pi
iii) f(x) = x * sin(x^2)
Using these information, this is what the integral would be the following:
To compute this integral, we can use a u-substitution setting x^2 as u.
u = x^2
du = 2x dx.
u(pi) = pi^2
u(pi/2) = (pi^2)/4
Hence, the integral would look like this in terms of u:
As the antiderivative is -cos(u), the integral would give the final value as -1/2 cos(pi^2) + 1/2 cos((pi^2)/4)
Hence, the final answer would be -1/2 cos(pi^2) + 1/2 cos((pi^2)/4).
Hi,
Thank you for your question.
This sigma notation has a form of riemann sums; hence, we can convert this sigma notation into an integral. From this sigma notation, we can get some key informations.
i) a = pi/2
ii) b = pi/2 + pi/2 = pi
iii) f(x) = x * sin(x^2)
Using these information, this is what the integral would be the following:
To compute this integral, we can use a u-substitution setting x^2 as u.
u = x^2
du = 2x dx.
u(pi) = pi^2
u(pi/2) = (pi^2)/4
Hence, the integral would look like this in terms of u:
As the antiderivative is -cos(u), the integral would give the final value as -1/2 cos(pi^2) + 1/2 cos((pi^2)/4)
Hence, the final answer would be -1/2 cos(pi^2) + 1/2 cos((pi^2)/4).
Does this answer your question?