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wextl, 12. ay, Sol. Here degree of numerator is greater than degree of denominator., Therefore, dividing the numerator by the denominator,, , xv—-1)xs84x41, , ad -x, = +, 2x +1, e4xt) Qx+1 ®, —_,J = oA., x21 x?-1, MathonGo 56, Class 12 Chapter 7 - Integrals, , , , , , , , , , , , , , | Rational function = Quotient + Remainder, Divisor, 2x +1 2x+1 A B si, let “ay = @+D@-) = x+1* x-1 @), Multiplying by L.C.M. = (x + 1)(x — 1), we have, 2x +1 = A(x - 1) + Ba + 1), or 2x+1=Ax-A+Bx+B, By equating the coefficients of x and constant terms, we get, A+B=2 tii), and ears .(iv), 3, (iii) + (iv) gives 2B=3 > Be 2, . 3. 3 1, Putting B = 2 in (iii), we gt A+ — =2 or A= 2, Putting values of A and B in eqn. (ii), we have, 1 3, 2x? an, x?-1 «“x+1 <x-1, = +41, Putting this value of 2-1 4 in (i),, - il 3, e+e+1 2 2, er 1 fee + 1 ee, 3, x txt) 1 1 3 1, j 1 de=fxde+5 eH dx + sls dx, , 3, +5 log | x+11+ 5 log|x-1]+e.