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Answer :
Final answer:
To evaluate the integral of 4x⁹e⁻ˣ⁵, we use integration by substitution with u = -x⁵, and eventually arrive at the final answer -4/5 e⁻ˣ⁵ + C, with C representing the constant of integration.
Explanation:
The question asks to evaluate the integral of 4x⁹e⁻ˣ⁵. To solve this, we can use the method of integration by substitution. Let's use u = -x⁵ as our substitution, which means that du = -5x⁴ dx. We also adjust our integral to match the substitution, which will change 4x⁹ to 4x⁵(-1/5)(-x⁴), simplifying the integral to -4/5 ∫eᵘ du. The antiderivative of -4/5 ∫eᵘ du is -4/5 eᵘ + C, where C is the constant of integration. Substituting back for u, we get -4/5 e⁻ˣ⁵ + C as the final answer. This is an example of an integral most easily evaluated by converting the function into an exponential with e as the base.
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