Solution:
Beginas usualby letting and Next,createa tablethat consistsof threecolumns,
as shownbelow:
Alternatesigns tivesand its deriva- and its antiderivatives-
To find the solutionfor the integral,pick the sign from the first row multiplyit by of the first row
and then multiplyby the of the secondrow, (watchthe directionof the arrows.)This is
the first term in the solution.Do the samethingto obtainthe secondterm:Pick the sign from the second
row, multiplyit by the of the samerow and then followthe arrowto multiplythe productby the in the
third row. Eventuallywe obtainthe solution
Solvingfor an UnknownIntegral
Thereare someintegralsthat requireus to evaluatetwo integrationsby parts,followedby solvingfor the
unknownintegral.Thesekindsof integralscrop up oftenin electricalengineeringand otherdisciplines.
Example6:
Evaluate.
Solution:
Let and Then and
Noticethat the secondintegrallooksthe sameas our originalintegralin form,exceptthat it has a in-
steadof To evaluateit, we againapplyintegrationby partsto the secondtermwith
and
Then