Mathematical Methods for Physics and Engineering - Matematica.NET

5.13 EXERCISESconstant limits of integration the order of integration and differentiation can bereversed.In the more general case where the limits of the integral are themselves functionsof x, it follows immediately thatI(x) =∫ t=v(x)t=u(x)which yields the partial derivatives∂I∂I= f(x, v(x)),∂vConsequentlyf(x, t) dt= F(x, v(x)) − F(x, u(x)),( ) ( )dI ∂I dv ∂I dudx = ∂v dx + ∂u dx + ∂I∂x= −f(x, u(x)).∂u= f(x, v(x)) dvdu− f(x, u(x))dx dx + ∂∂x= f(x, v(x)) dv∫du v(x)− f(x, u(x))dx dx + u(x)∫ v(x)u(x)f(x, t)dt∂f(x, t)∂xdt, (5.47)where the partial derivative with respect to x in the last term has been takeninside the integral sign using (5.46). This procedure is valid because u(x) andv(x)are being held constant in this term.◮Find the derivative with respect to x of the integralApplying (5.47), we see thatdIdxI(x) =∫ x 2sin x3 sin x2= (2x) −x 2= 2sinx3x−xsin x2xsin xtdt.t∫ x 2x (1) + [ sin xt+xsin x3 sin x2=3 − 2x x= 1 x (3 sin x3 − 2sinx 2 ). ◭x] x 2xt cos xtdtt5.13 Exercises5.1 Using the appropriate properties of ordinary derivatives, perform the following.179