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प्रश्न
Evaluate the following integral:
`sqrt(9x^2 + 12x + 3) "d"x`
उत्तर
`sqrt(9x^2 + 12x + 3) "d"x = int sqrt(9(x^2 + 12/9x + 3/9)) "d"x`
= `3int sqrt(x^2 + 4/3x + 1/3 d"x`
= `3sqrt((x + 4/6)^2 - 16/36 + 1/3) "d"x`
= `3int sqrt((x + 2/3)^2 - 1/9) "d"x`
= `3 int sqrt((x + 2/3)^2 - (1/3)^2) "d"x`
= `3[((x + 2/3))/2 sqrt((x + 2/3)^2 - 1/9) - 1/((9)(2)) log |(x + 2/3) + sqrt((x + 2/3)^2 - 1/9)|] + "c"`
= `3[(3x + 2)/6 sqrt((3x + 2)^2/9 - 1/9) - 1/18 log|((3x + 2))/2 + sqrt((3x + 2/9)^2 - 1/9)|] + "c"`
= `3[((3x + 2))/6 sqrt(9x^2 + 12x + 3)/3 - 1/18 log |((3x + 2))/3 + sqrt(9x^2 + 12x + 3)/3|] + "c"`
= `((3x - 2))/6 sqrt(9x^2 + 12x + 3) - 1/6 log |(3x + 2) + sqrt(9x^2 + 12x + 3)| + "k"`
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