Dear student Let I = ∫(x-4)ex(x-2)3dxPut x-2 = t⇒dx = dtNow, I =∫t-2et+2t3 dt=e2∫t-2 ett3 dt=e2∫ett2 dt - ∫2ett3 dt=e21t2×et - ∫-2t3×etdt - ∫2ett3 dt=e2ett2 + ∫2ett3 dt - ∫2ett3 dt=et+2t2 + C=exx-22 + C Regards

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Dear student

Let I = \int \frac{(x - 4) e^{x}}{(x - 2)^{3}} dx Put (x - 2) = t \Rightarrow dx = dt Now, I = \int \frac{(t - 2) e^{t + 2}}{t^{3}} dt = e^{2} \int \frac{(t - 2) . e^{t}}{t^{3}} dt = e^{2} [\int \frac{e^{t}}{t^{2}} dt - \int \frac{2 e^{t}}{t^{3}} dt] = e^{2} [\frac{1}{t^{2}} \times e^{t} - \int (- \frac{2}{t^{3}} \times e^{t}) dt - \int \frac{2 e^{t}}{t^{3}} dt] = e^{2} [\frac{e^{t}}{t^{2}} + \int \frac{2 e^{t}}{t^{3}} dt - \int \frac{2 e^{t}}{t^{3}} dt] = \frac{e^{t + 2}}{t^{2}} + C = \frac{e^{x}}{{(x - 2)}^{2}} + C

Regards

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