Q.
∫1−x2x(1−2x)dx is equal to
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a
x+logx−34log|x−1|+C
b
x2+logx+14log|x−1|+C
c
x2+logx−34log|2x−1|+C
d
None of the above
answer is C.
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Detailed Solution
Let I=∫1−x2x(1−2x)dxHere, degree of numerator is equal to degree of denominator, so divide the numerator by denominator.Thus,1−x2x(1−2x)=x2−12x2−x=12+12x−12x2−x∴ I=∫12dx+∫12x−12x2−xdx⇒I=I1+I2 .....(i)Where, I1=∫12dx and I2=∫12x−12x2−xdxNow, I1=∫12dx=12x+C1 and I2=∫12x−12x2−xdxLet 12x−12x2−x=Ax+B(2x−1)⇒12x−12x2−x=A(2x−1)+Bxx(2x−1)⇒ 12x−1=2Ax−A+Bx⇒ 12x−1=x(2A+B)−AOn comparing the coefficient of x and constant term on both sides, we get 2A+B=12 .....(ii) and −A=−1⇒A=1 .....(iii) From Eq. (iii), 2×1+B=12⇒ B=12−2=−32∴ I2=∫1x−32(2x−1)dx =∫1xdx−32∫12x−1dx⇒ I2=logx−32log|2x−1|2+C2On putting the values of l1 and I2 in Eq. (i), we getI=12x+logx−34log|2x−1|+C C1+C2=C
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