Final answer:
The integral of t⁴eᴇ⁵dt is found using integration by parts, where u=t⁴ and dv=eᴇ⁵ dt, and the process is repeated until all terms of t are exhausted.
Step-by-step explanation:
The integral of t⁴eᴇ⁵dt can be found using the technique of integration by parts. The formula for integration by parts is ∫ u dv = uv - ∫ v du, where u and dv are parts of the original integrand. Choose u=t⁴, which implies du=4t³ dt, and dv=eᴇ⁵ dt, which implies v=(1/5)eᴇ⁵ when differentiated with respect to t. Applying integration by parts, we get:
- ∫ t⁴ eᴇ⁵ dt = t⁴ * (1/5)eᴇ⁵ - ∫ (1/5)eᴇ⁵ * 4t³ dt
- Continue this process with the new integral until all terms of t are exhausted.
- Simplify and combine all terms to obtain the final answer. To check the integration, differentiate the result and it should match the original integrand t⁴eᴇ⁵.
Remember that the constant of integration should be included in the final result.
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