InterviewSolution
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InterviewSolution
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Evaluate `int((2x+1))/((4-3x-x^(2)))dx`. |
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Answer» Let `(2x+1)=A*(d)/(dx)(4-3x-x^(2))+B`. Then , `(2x+1)=A(-3-2x)+B` . . . (i) Comparing the coefficients of like terms , we get `(-2A=2and-3A+B=1)rArr(A=-1,B=-2)`. `thereforeint((2x+1))/((4-3x-x^(2)))dx = int{((-1)*(-3-2x)-2)/((4-3x-x^(2)))}dx` `=-int((3-2x))/((4-3x-x^(2)))dx-2int(dx)/((4-3x-x^(2)))` `=-log|4-3x-x^(2)|+2int(dx)/((4-3x-x^(2)))` `=-log|4-3x^(2)-x^(2)|+2int(dx)/((x+(2)/(2))^(2)-(4+(9)/(4)))` `=-log|4-3x-x^(2)|+2int(dx)/({(x+(3)/(2))^(2)-((5)/(2))^(2)})` `=-log|4-3x-x^(2)|+(2)/((2xx(5)/(2)))log|((x+(3)/(2))-(5)/(2))/((x+(3)/(2))+(5)/(2))|+C` `=-log|4-3x-x^(2)|+(2)/(5)log|(x-1)/(x+4)|+C`. |
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