Compare the results. Antiderivatives of xe^4x and xe^5x Solution You need to integrate the functions [y = xe^(4x)] and [y = xe^(5x)] using parts such that: [int xe^(4x) dx = x*(e^(4x))/4 - int (e^(4x))/4 dx ] [f(x) = x => f\'(x) = 1] [g\'(x) = e^(4x) => g(x) = (e^(4x))/4] [int xe^(4x) dx = x*(e^(4x))/4 - (e^(4x))/16 + c] Reasoning by analogy yields: [int xe^(5x) dx = x*(e^(5x))/5 - (e^(5x))/25 dx + c] Hence, evaluating the antiderivatives of the given functions yields [int xe^(4x) dx = x*(e^(4x))/4 - (e^(4x))/16 + c] and [int xe^(5x) dx = x*(e^(5x))/5 - (e^(5x))/25 dx + c] ..

1. Compare the results. Antiderivatives of xe^4x and xe^5x
Solution
You need to integrate the functions [y = xe^(4x)] and [y = xe^(5x)] using parts such that:
[int xe^(4x) dx = x*(e^(4x))/4 - int (e^(4x))/4 dx ]
[f(x) = x => f'(x) = 1]
[g'(x) = e^(4x) => g(x) = (e^(4x))/4]
[int xe^(4x) dx = x*(e^(4x))/4 - (e^(4x))/16 + c]
Reasoning by analogy yields:
[int xe^(5x) dx = x*(e^(5x))/5 - (e^(5x))/25 dx + c]
Hence, evaluating the antiderivatives of the given functions yields [int xe^(4x) dx = x*(e^(4x))/4
- (e^(4x))/16 + c] and [int xe^(5x) dx = x*(e^(5x))/5 - (e^(5x))/25 dx + c] .