Question

find integral x³cos(x³+2)dx​

Answer 1

Answer:

Split the integrand in two items:

F = 1/2 Sqrt[4 - x^2] + x^3 Cos[x/2] Sqrt [4-x^2]

The second item is an odd function, and its integral within [-2, +2] is zero.

Sqrt [4-x^2] is a half-circle with center at the origin and radius 2. The area of the half-circle is 2 Pi, so Integral [F] = Pi