Mathematical Methods for Physics and Engineering - Matematica.NET

PRELIMINARY CALCULUSSincethe required relationship isdtdx = 1 x 2 sec2 2 = 1 (1+tan 2 x )= 1+t2 ,2 2 2dx = 2 dt. (2.34)1+t2 ◮Evaluate the integral∫I =21+3cosx dx.Rewriting cos x in terms of t and using (2.34) yields∫I =∫=∫=∫=21+3 [ (1 − t 2 )(1 + t 2 ) −1] ( 21+t 2 )dt2(1 + t 2 )1+t 2 +3(1− t 2 )∫22 − t dt = 2( 21+t 2 )dt2( √ 2 − t)( √ 2+t) dt(1 1√ √ + 1 )√ dt2 2 − t 2+t= − 1 √2ln( √ 2 − t)+1 √2ln( √ 2+t)+c[= √ 1√ ]2+tan(x/2)ln √ + c. ◭2 2 − tan (x/2)Integrals of a similar form to (2.33), but involving sin 2x, cos2x, tan 2x, sin 2 x,cos 2 x or tan 2 x instead of cos x and sin x, should be evaluated by using thesubstitution t =tanx. Inthiscasesin x =t√1+t2 , cos x = 1√1+t2and dx = dt1+t 2 . (2.35)A final example of the evaluation of integrals using substitution is the methodof completing the square (cf. subsection 1.7.3).66