Final answer:
To solve the given integral, we can rewrite it as I = A ∫ₐ ᵇ u e^-u du. Let's substitute u = x⁵ and use integration by parts to evaluate the integral. The final result is I = 5A (-e^(-u) u + e^(-u) + C).
Step-by-step explanation:
To solve the given integral, we can rewrite it as I = A ∫ₐ ᵇ u e^-u du. Let's substitute u = x⁵. Then, we have du = 5x⁴ dx. Also, the limits of integration change to a = 1⁵ and b = 3⁵. Substituting these values, we get I = 5A ∫₁³ u e^(-u) du.
Now, let's evaluate the integral ∫₁³ u e^(-u) du. This can be done using integration by parts. Let's set f(u) = u and g'(u) = e^(-u). By applying the integration by parts formula, we get ∫ u e^(-u) du = -e^(-u) u - ∫ -e^(-u) du. Simplifying this, we obtain ∫ u e^(-u) du = -e^(-u) u + e^(-u) + C, where C is the constant of integration. Therefore, the final result is I = 5A (-e^(-u) u + e^(-u) + C).