Use the method of integration by parts to determine the primitives for this function:?

The above answer is wrong because exponents come before multiplication by default. There is no need to use integration by parts for this integral.

Integrate this expression using the power rule:

∫ x²℮²x dx = ℮² ∫ x³ dx

∫ x²℮²x dx = ℮²x⁴ / 4 + C

fundamental of fg' = f g - fundamental of f' g the factor to observe is that f could be actual differentiable and g actual integrable, so your chooice of f and g is okay: f = a million - (a million/2) t , so f' = -a million/2 g' = cos (mt), so g = (a million/m) sin (mt) and f'g, being some consistent distinctive of sin (mt), may be readity built-in...

u = x², du = 2x dx

dv = e²˟ dx , v = (1/2)e²˟

ʃ udv = uv - ʃ vdu = x²(1/2)e²˟ - ʃ(2x)(1/2)e²˟du