InterviewSolution
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InterviewSolution
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Evaluate: `int(5x-2)/(1+2x+3x^2)dx` |
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Answer» Let `(5x-2)=A*(d)/(dx)(1+2x+3x^(2))+B`. Then , `(5x-2)=A(6x+2)+B`. " " ...(i) Comparing the coefficients of like powers of x on both sides , we get `6A=5and2A+B=-2` This gives `A=(5)/(6)andB=(-11)/(3)`. `thereforeI=int({(5)/(6)(6x+2)-(11)/(3)})/((1+2x+3x^(2)))dx` `=(5)/(6)*int(6x+2)/((1+2x+3x^(2)))dx-(11)/(3)int(dx)/((3x^(2)+2x+1))` `=(5)/(6)log|1+2x+3x^(2)|-(11)/(9)*int(dx)/({(x+(1)/(3))^(2)+((1)/(3)-(1)/(9))})` `=(5)/(6)log|1+2x+3x^(2)|-(11)/(9)*int(dx)/({(x+(1)/(3))^(2)+((sqrt(2))/(3))^(2)})+C` `=(5)/(6)log|1+2x+3x^(2)|-(11)/(9)*(1)/(((sqrt(2))/(3)))tan^(-1){(x+(1)/(3))/(((sqrt(2))/(3)))}+C` `=(5)/(6)log|1+2x+3x^(2)|-(11)/(3sqrt(2))tan^(-1)((3x+1)/(sqrt(2)))+C`. |
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