InterviewSolution
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InterviewSolution
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`int((5x-2))/((1+2x+3x^(2)))dx` का मान ज्ञात कीजिए । |
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Answer» `int((5x-2))/((1+2x+3x^(2)))dx` अब माना कि `(5x-1)=A.(d)/(dx)(1+2x+3x^(2))+B` तब `" "(5x-2)=A(6x+2)+B" …(1)"` अब, समीकरण (1 ) के दोनों पक्षों के x की घातों की तुलना करने पर `6A=5" तथा "2A+B=-2` `rArr" "A=(5)/(6)" तथा "B=(-11)/(3)` तब `" "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)/(3).(1)/(3)int(dx)/((x^(2)+(2)/(3)x+(1)/(3)))` `=(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)+((sqrt2)/(3))^(2)})+c` `=(5)/(6)log(1+2x+3x^(2))-(11)/(9).(1)/(((sqrt2)/(3)))tan^(-1){(x+(1)/(3))/((sqrt2)/(3))}+c` `=(5)/(6)log(1+2x+3x^(2))-(11)/(3sqrt2)tan^(-1)((3x+1)/(sqrt2))+c` |
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