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2, , 18. G@=x)(+27), , , , 2, Sol. To find integral of the Rational function G-n0+22)°, 2 A Be +C, Let int (Ct -eeeeeee Sanna aloe ee ca, et integran a ~d+ 2) I + le (i), , (Partial Fractions), Multiplying by L.C.M. = (1 — x)(1 + x?), 2 = A(1 + x”) + (Bx + C)(1 - x), , or 2 = A+ Ax? + Bx — Bx? + C - Cx, Comparing coefficients of x?, x and constant terms, we have, , x? A-B=0 (ti), x B-C=0 «(iii), Constant terms A + C = 2 «(iv), , Let us solve (ii), (iii), (iv) for A, B, C, From (ii), A = B and from (iii), B=C, , , , MathonGo 57, Class 12 Chapter 7 - Integrals, , a A=Be=C, , Putting A = C in (iv), C+ C=2 or 2C =2 or C=l1, , « A=Ce=l1 « BzeAz=l, , Putting these values of A, B, C in (i),, , 2 _ 1 x+1 1 x 1, (L-x)1+x*) l-x " l+x2 l-x ° 1l+x7 ° 1442, , 1 1 2x 1, , , , , , , , , , : 2 _ 1 1 Qx 1, Nios * IE d+ 5] z dx+f ae, , , , , , x ex,, log |1- x1 1, = SloGumimians * 5 log | 1+ x7| + tan-+x+c, of 25 dx =f Fe) d= Wg sco, , i, =—log|1-x|+ 5 log (+x) + tam tate, , (14+? > 0, therefore | 1 + x? | = 1 + x?), Note. log | 1 — x | = log | - (x - 1) |, = log | x — 1| because | —¢ | =| ¢ |.