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# 2 in the particular case in which ac b 2 the integral

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2. In the particular case in which ac = b 2 the integral is - D a ( ax + b ) + A a log | ax + b | . 3. Show that if the roots of Q ( x ) = 0 are all real and distinct, and P ( x ) is of lower degree than Q ( x ), then Z R ( x ) dx = P ( α ) Q 0 ( α ) log | x - α | , the summation applying to all the roots α of Q ( x ) = 0. [The form of the fraction corresponding to α may be deduced from the facts that Q ( x ) x - α Q 0 ( α ) , ( x - α ) R ( x ) P ( α ) Q 0 ( α ) , as x α .] 4. If all the roots of Q ( x ) are real and α is a double root, the other roots being simple roots, and P ( x ) is of lower degree than Q ( x ), then the integral is A/ ( x - α ) + A 0 log | x - α | + B log | x - β | , where A = - 2 P ( α ) Q 00 ( α ) , A 0 = 2 { 3 P 0 ( α ) Q 00 ( α ) - P ( a ) Q 000 ( α ) } 3 { Q 00 ( α ) } 2 , B = P ( β ) Q 0 ( β ) , and the summation applies to all roots β of Q ( x ) = 0 other than α . 5. Calculate Z dx { ( x - 1)( x 2 + 1) } 2 .

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[VI : 132] DERIVATIVES AND INTEGRALS 285 [The expression in partial fractions is 1 4( x - 1) 2 - 1 2( x - 1) - i 8( x - i ) 2 + 2 - i 8( x - i ) + i 8( x + i ) 2 + 2 + i 8( x + i ) , and the integral is - 1 4( x - 1) - 1 4( x 2 + 1) - 1 2 log | x - 1 | + 1 4 log( x 2 + 1) + 1 4 arc tan x. ] 6. Integrate x ( x - a )( x - b )( x - c ) , x ( x - a ) 2 ( x - b ) , x ( x - a ) 2 ( x - b ) 2 , x ( x - a ) 3 , x ( x 2 + a 2 )( x 2 + b 2 ) , x 2 ( x 2 + a 2 )( x 2 + b ) 2 , x 2 - a 2 x 2 ( x 2 + a 2 ) , x 2 - a 2 x ( x 2 + a 2 ) 2 . 7. Prove the formulae: Z dx 1 + x 4 = 1 4 2 log 1 + x 2 + x 2 1 - x 2 + x 2 + 2 arc tan x 2 1 - x 2 , Z x 2 dx 1 + x 4 = 1 4 2 - log 1 + x 2 + x 2 1 - x 2 + x 2 + 2 arc tan x 2 1 - x 2 , Z dx 1 + x 2 + x 4 = 1 4 3 3 log 1 + x + x 2 1 - x + x 2 + 2 arc tan x 3 1 - x 2 . 131. Note on the practical integration of rational functions. The analysis of § 130 gives us a general method by which we can find the integral of any real rational function R ( x ), provided we can solve the equation Q ( x ) = 0. In simple cases (as in Ex. 5 above) the application of the method is fairly simple. In more complicated cases the labour involved is sometimes prohibitive, and other devices have to be used. It is not part of the purpose of this book to go into practical problems of integration in detail. The reader who desires fuller information may be referred to Goursat’s Cours d’Analyse , second ed., vol. i, pp. 246 et seq. , Bertrand’s Calcul Int´ egral , and Dr Bromwich’s tract Elementary Integrals (Bowes and Bowes, 1911).
[VI : 133] DERIVATIVES AND INTEGRALS 286 If the equation Q ( x ) = 0 cannot be solved algebraically, then the method of partial fractions naturally fails and recourse must be had to other methods. * 132. Algebraical Functions. We naturally pass on next to the question of the integration of algebraical functions. We have to consider the problem of integrating y , where y is an algebraical function of x . It is however convenient to consider an apparently more general integral, viz. Z R ( x, y ) dx, where R ( x, y ) is any rational function of x and y . The greater generality of this form is only apparent, since ( Ex. xiv . 6) the function R ( x, y ) is itself an algebraical function of x . The choice of this form is in fact dictated simply by motives of convenience: such a function as px + q + ax 2 + 2 bx + c px + q - ax 2 + 2 bx + c is far more conveniently regarded as a rational function of x and the simple algebraical function ax 2 + 2 bx + c , than directly as itself an algebraical

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