Can someone help me fully breakdown this problem Thank You W

Can someone help me fully breakdown this problem? Thank You

Write the first three nonzero terms of the Maclaurin series of sin(x). Write the first three nonzero terms of the Maclaurin series of sin(2x2). Estimate the value of the definite integral sin(2x2)dx, expressing the estimate as a sum of two fractions. Give an upper bound for the error of your estimate with an explanation of how the error bound was obtained.

Solution

x-1/6x^3+1/120x^5.... The series goes something like this.. f(x) + f\'(x)x + f\'\'(x)x^2/2! + f\'\'\'(x)x^3/3! ...... f(x) = sin2x^2 f\'(x) = 4sin2xcos2x f\'\'(x) = 4(2cos2x^2 + 2sin2x^2) = 8(2) = 16 Okay, someting like that..And once you find the deravatives, you plug it into the give Formula f(x) + f\'(x)x + f\'\'(x)x^2/2! + f\'\'\'(x)x^3/3! ......