InterviewSolution
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InterviewSolution
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`int(3x+1)/(2x^(2)-2x+3)dx` का मान ज्ञात कीजिए । |
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Answer» माना `3x+1=A.(d)/(dx)(2x^(2)-2x+3)+B` `rArr" "3x+1=A(4x-2)+B` x के गुणांकों की तथा अचर पदों की तुलना करने पर `{:(,3=4A," और ",1=-2A+B),(rArr.,A=(3)/(4)," और ",B=(5)/(2)):}` `therefore int(3x+1)/(2x^(2)-2x+3)dx=int((3)/(4)(4x-2)+(5)/(2))/(2x^(2)-2x+3)dx` `" "=(3)/(4)int(4x-2)/(2x^(2)-2x+3)dx+(5)/(2)int(1)/(2x^(2)-2x+3)dx` `=(3)/(4)int(1)/(t)dt+(5)/(4)int(1)/(x^(2)-x+(3)/(2))dx` माना `" "2x^(2)-2x+3=t` `" "(4x-2)dx=dt` `=(3)/(4)log|t|+(5)/(4)int(1)/((x^(2)-x+(1)/(4))+((3)/(2)-(1)/(4)))dx` `=(3)/(4)log|2x^(2)-2x+3|+(5)/(4) int(1)/((x-(1)/(2))^(2)+((sqrt5)/(2))^(2))dx` `=(3)/(4)log|2x^(2)-2x+3|+(5)/(4).(1)/(sqrt5//2)tan^(-1).(x-(1)/(2))/((sqrt5)/(2))+c` `=(3)/(4)log|2x^(2)-2x+3|+(sqrt5)/(2)tan^(-1)((2x-1)/(sqrt5))+c` |
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