Download PDF by William Squire: Integration for Engineers and Scientists (Modern analytic

By William Squire

ISBN-10: 0444000755

ISBN-13: 9780444000750

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Additional info for Integration for Engineers and Scientists (Modern analytic and computational methods in science and mathematics)

Example text

61b) a - t2 It is obvious in both cases that y and dy/dt are rational functions of t. These transformations are special cases of a general principle: The integral of a rational function of x, YI , Y2, .. , y,, can be reduced to the integral of a rational function of t if there is a parametric representation x = Ro(t) Y, = R, (t) y2 = R2(t) y = R,, (t) where the R's are rational functions.

63) a The individual inequalities are relatively easy to prove. 61) is derived by noting that 1(A) = f". 64) must be positive for all real values of A. 65) where Al and J2 are the roots of the quadratic I(A) = 0. (A) is negative. The condition for the roots of I(A) to be equal or complex is that the discriminant 2 2 fgdx -4 fZdx ('ab { ,l Transposing gives Eq. 61). g2 dx ,0. 1 The Chebyshev Inequality A different type of inequality was first stated by Chebyshev. 67) when y < x. 69) [u(X)-u(y)][v(x)-v(y)]dxdy is positive if u and v are the same type and negative if they are different.

2 (a) Show that J R, (x) In x dx is elementary if R, (x) = (c/x) + R2 (x) where R, and R2 are rational functions. (b) Show that f (In x)l(x-a) dx is only elementary when a = 0. (c) Show that lnx (x-a)(x-b) dx is only elementary when a = b (Hardy). 3 (a) Show that ! eR, (x) dx can only involve e'R2 (x) + E ax 5 e'/(x- f k) dx and is only elementary when all the xk's vanish. (b) Obtain analogous results for 5 sin x R,(x)dr and J cos x R,(x)dx (Hardy). 4) 4 49 ANALYTIC EVALUATION OF INDEFINITE INTEGRALS Investigate whether the following are elementary.

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Integration for Engineers and Scientists (Modern analytic and computational methods in science and mathematics) by William Squire


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