Question
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\int (e^{x} - 8)^{5} e^{x} \,dx = \frac{(e^{x} - 8)^{6}}{6} + C
\]
Solution
\[
\int \left(e^{x} - 8\right)^{5} e^{x} \,dx = \frac{\left(e^{x} - 8\right)^{6}}{6} + C
\]
**Explicación:**
Para resolver la integral \(\int (e^{x} - 8)^{5} e^{x} \,dx\), realizamos un cambio de variable:
Sea \( u = e^{x} - 8 \), entonces \( du = e^{x} dx \).
Sustituyendo en la integral:
\[
\int u^{5} \, du = \frac{u^{6}}{6} + C = \frac{(e^{x} - 8)^{6}}{6} + C
\]
donde \( C \) es la constante de integración.
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Simplify this solution Extra Insights
To solve the integral \( \int (e^x - 8)^5 e^x \, dx \), let’s use integration by substitution. Let \( u = e^x - 8 \). Then, the differential \( du = e^x \, dx \). This transforms the integral into: \[ \int u^5 \, du \] Now, integrating \( u^5 \) gives us: \[ \frac{u^6}{6} + C \] Substituting back for \( u \): \[ \frac{(e^x - 8)^6}{6} + C \] So, the final answer is: \[ \int (e^x - 8)^5 e^x \, dx = \frac{(e^x - 8)^6}{6} + C \]
