SOLUTIONNotice thatt2becomes simpler when differentiated (wherease5tis mostly unchanged when differentiated or integrated), so we choose Thendu=u=dv=dtThendu=dtv= Find t^2e^5tdt. SOLUTION Notice that t^2becomes simpler when differentiated (whereas e^5t is mostly unchanged when differentiated or integrated), so we choose Integration by parts gives The integral that we obtained, te^5t dt, is simpler than the original integral but is still not obvious. Therefore, we use integration by parts a second time, this time withu=tanddv=e^5tdt. Thendu= Putting this in Equation 3, we get Solution t*T e^(5t)/5 - (2/5)[ te^(5t)/5 - e^(5t)/25 ] t*t e^(5t)/5 - (2/25) te^(5t) + 2/125 e^(5t) ].