Example 3 Video Example () Find theat at . Solution Notice that t2 becomes simpler when differentiated (whereas e is mostly unchanged when differentiated or integrated), so we choose u = t2 and dv = e4t dt. Then du = dt and v = Integration by parts gives the At at = - teat at ( 3 ) . The integral that we obtained, te4t dt, is simpler than the original integral but is still not obvious. Therefore, we use integration by parts a second time, this time with u = t and dv = e dt. Then du = dt, v = and / tett at = = tett - dt = + C. Putting this in Equation (3), we get the4t at = _theft te 4t dt 4t )( treat -. 16 " + c ) + C1, where C1 = - LC. 2Example 2 Video Example () Evaluate / In(7x) dx. Solution Here we don't have much choice for u and dv. Let u = In(7x) and dv = dx. Then du = dx and v = Integrating by parts, we get In ( 7x) dx = In ( 7x ) - x. dx In ( 7 x ) - dx + C. Integration by parts is effective in this example because the derivative of the function f(x) = In(7x) is simpler than f. Need Help? Read It