Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 291 ANTIDERIVATIVES 223EXAMPLENotice that we find a specific antiderivative, x3/3 + x4/4, and then add the “arbitrary” constant C.Rules 1 through 4 enable us to compute the antiderivative of any polynomial.1 1 xs2 5EXAMPLE J(3x5 -2 x4 + 7x2 + x - 3) dx = 3 ( 3 - -_-_--x6 x5 7 x22 10 3 2(-) + 7 ( 3 + 2-+ - x3 + - - 3x + c2 3x + cThe next rule will prove to be extremely useful.s(g(x))’+ +RULE 5 (Quick Formula I). (g(x))’g’(x)dx = r-flThe power chain rule implies thatcwhich yields quick formula I.EXAMPLESJ(; x2 + 5)7x dx =; (; xz + 5)’+ c1{J- dx = 5 I(2x - 5)’12(2) dx = -1 (2x - 5)3’2 12 3 3+ c = - (2x - 5)3’z + cRULE 6 (Substitution Method). Deferring the general formulation and justification to Problem 29.18,we illustrate the method by three examples.(i) Find x2 cos x3 dx. Let x3 = U. Then, by Section 21.3, the differential of u is given byNow substitute U for x3 and idu for x2 dx,s s: 31du = Dx(x3) dx = 3x2 dx or x2 dx = - du3x2 cos x3 dx = - cos u du =1 13cos u du = - sin u + c = - sin x3 + C