# Using the method of integration for parts show that ∫xe^(x/2) dx = 2xe^(x/2) - 4e^(x/2) + constant You need to remember the formula of integration by parts such that:

int udv = uv - int vdu

Selecting u = x  and dv = e^(x/2)dx  yields:

u = x => du = dx

dv = e^(x/2)dx => v = (e^(x/2))/(1/2)

Substituting dx  for du  and (e^(x/2))/(1/2) ...

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You need to remember the formula of integration by parts such that:

int udv = uv - int vdu

Selecting u = x  and dv = e^(x/2)dx  yields:

u = x => du = dx

dv = e^(x/2)dx => v = (e^(x/2))/(1/2)

Substituting dx  for du  and (e^(x/2))/(1/2)  for v yields:

int xe^(x/2)dx = x(e^(x/2))/(1/2) - int (e^(x/2))/(1/2)dx

int xe^(x/2)dx = 2x(e^(x/2)) - 2(e^(x/2))/(1/2) + c

int xe^(x/2)dx = 2x(e^(x/2)) - 4(e^(x/2)) + c

Hence, evaluating the given integral using parts yields int xe^(x/2)dx = 2x(e^(x/2)) - 4(e^(x/2)) + c .

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