Use substitution to find an indefinite integral with exponential functions

Question

∫4t^4e^2t^5

Please explain each step in detail. Thank you!

Answer 1

Since we have the integrand multiplied by t to a power one less than the power in the exponent on e (4 vs. 5), we can solve this problem by a simple u-substitution. Let u = 2*(t^5), then du = 10*(t^4) dt and the integral transforms into ∫ (2/5)*e^u du. So the answer is (2/5)*e^u + C, and we perform the reverse substitution u = 2*(t^5) to get the answer in terms of our original variable, t. Therefore, the answer to the problem is (2/5)*e^(2*(t^5)) + C.