# integral(x^2/(x^2+7 x+10), x)

sciencesolve | Teacher | (Level 3) Educator Emeritus

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`int (x^2)/(x^2+7x+10) dx = int (x^2+7x+10)/(x^2+7x+10) dx - int (7x+10)/(x^2+7x+10) dx`

`int (x^2)/(x^2+7x+10) dx = int dx - int (7x+10)/(x^2+7x+10) dx`

You should use partial fraction decomposition to evaluate the integral `int (7x+10)/(x^2+7x+10) dx`  such that:

`(7x+10)/(x^2+7x+10) = (7x+10)/((x+2)(x+5))`  (you should write denominator in factored form)

`(7x+10)/((x+2)(x+5)) = a/(x+2) + b/(x+5)`

`7x+10 = x(a+b) + 5a + 2b`

Equating the coefficients of like terms yields:

`a+b = 7 =gt a = 7 - b`

`5a + 2b = 10 =gt 5(7-b) + 2b = 10`

`35 - 5b + 2b = 10 =gt -3b = -25 =gt b = 25/3`

`a = 7 - 25/3 =gt a = -4/3`

`(7x+10)/((x+2)(x+5)) = -4/(3(x+2)) + 25/(3(x+5)) `

`int (7x+10)/(x^2+7x+10) dx = int-4/(3(x+2)) dx + int 25/(3(x+5)) dx`

`int (7x+10)/(x^2+7x+10) dx = -(4/3)ln|x+2| + (25/3)ln|x+5| + ` c

Hence, evaluating the given integral yields `int (x^2)/(x^2+7x+10) dx = x + (4/3)ln|x+2| - (25/3)ln|x+5| + c.`