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La integral es \( \frac{(e^{x} - 8)^{6}}{6} + C \).
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To solve the integral \( \int ( e^{x} - 8 )^{5} e^{x} \, dx \), we can use substitution. Let \( u = e^{x} - 8 \). Then, the differential \( du = e^{x} \, dx \). Now, substitute into the integral: \[ \int (u^{5}) \, du \] This integral becomes: \[ \frac{u^{6}}{6} + C \] Substituting back for \( u \): \[ \frac{(e^{x} - 8)^{6}}{6} + C \] Thus, the final answer is: \[ \int ( e^{x} - 8 )^{5} e^{x} \, dx = \frac{(e^{x} - 8)^{6}}{6} + C \]
