No links please Q Evaluate : ∫x-16x2+x-1dx (ii) Ans : -1360ln6x-26x+3+112ln6x2+x-+C - Maths - Integrals

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Q Evaluate : ∫ x - 1 6 x 2 + x - 1 d x
(ii) Ans : - 13 60 ln 6 x - 2 6 x + 3 + 1 12 ln 6 x 2 + x - + C

Neha Sethi · Accepted Answer

Dear student
Let I=∫x-16x2+x-1dxWrite x-1 as 11212x+1+-1312 and splitI=∫12x+1126x2+x-1-13126x2+x-1dx=112∫12x+16x2+x-1dx-1312∫dx6x2+x-1Consider,112∫12x+16x2+x-1dxTake u=6x2+x-1⇒dx=112x+1duSo,112∫12x+16x2+x-1dx=112∫duu=112logu=112log6x2+x-1Consider, 1312∫dx6x2+x-1=1312∫1(2x+1)(3x-1)dxLet 1(2x+1)(3x-1)=A2x+1+B3x-11=A(3x-1)+B(2x+1)1=(3A+2B)x+(-A+B)On comparing, we get3A+2B=0 -A+B=1On solving, we getA=-25,B=35So,1312∫1(2x+1)(3x-1)dx=1312-25∫dx2x+1+35∫dx3x-1=1312-15×log(2x+1)+15log(3x-1)So,I=112log6x2+x-1-1312-15×log(2x+1)+15log(3x-1)+C=112log6x2+x-1-1360log3x-12x+1+CPlease recheck your final answer
Regards

No links please Q. Evaluate : ∫ x - 1 6 x 2 + x - 1 d x (ii) Ans : - 13 60 ln 6 x - 2 6 x + 3 + 1 12 ln 6 x 2 + x - + C

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Q. Evaluate : $\int \frac{(x - 1)}{6 x^{2} + x - 1} d x$
(ii) Ans : $- \frac{13}{60} \ln |\frac{6 x - 2}{6 x + 3}| + \frac{1}{12} \ln |6 x^{2} + x -| + C$