Advanced Calculus fi..

256 Advanced Calculus. Fifth Edition t ?Finally,a' - a l" f (x, w)dx =--awawby Leibnitz's Rule. Since w = t, dw/dt = 1 and the third term is accounted for.PROBLEMS1. Obtain the indicated derivatives in the form of integrals:a) $ 1: - dx d 2 x2 dxb) ai Jl (1- ,,,zC) & 1; log (xu) dx2. Obtain the indicated derivatives:a) & 1; t2dt b) ai d Jl t2 sin (x2)dxC) $~?log(l +x2)dx d) +d" 2 slnxd) &7 1, x--y dx4x jtanx x e-r2 dt3. Prove the following:d cvsaa) a(;; Ssina 1%Coscr +adx = 1% -[sin log (COS a+a) + cos a log (sin a+a)];b) $1: usinuxdx =O;C) &/:' e-'2~' dx = 2ye-~* - e-~4 - Zy J: x'e-"'~' dx.4. a) Evaluate 1; xn log x dx by differentiating both sides of the equation 1; xn dx =1with respect to n (n > - 1).b) Evaluate 1: ~"e-"~ dx by repeated differentiation of jr e-"" dx (a > 0).e) Evaluate (x2 yy,),, by repeated differentiation of jr x2:y2.[In (b) and (c) the improper integrals are of a type to which Leibnitz's Rule is applicable,as is shown in Chapter 6. The result of (a) can be explicitly verified.]5. Leibnitz's Rule extends to indefinite integrals in the form:J_ IThere is still an arbitrary constant in the equation-because we are evaluating an indejniteintegral. Thus from the equationone deduces that 1Jxetxdx=eu(5-$)+~I.a) By differentiating n times, prove thatdx (-1)n-1 an-1 i , .T-/(x2+ar (n-l)!aan-l8'' slnaxb) ~rovejx~cosaxdx = ;i;;"(7) + ~ , n=4,8, 12, .... 4