please help me with this integrate ∫3x^2+1/(x^2+x+2)^3
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You need to convert the denominator to the form `x^2+a^2` , hence, you need to complete the square using the formula `(a+b)^2 = a^2 + 2ab + b^2` , such that:
`x^2 + x = a^2 + 2ab =gt {(x^2=a^2 =gt x=a),(x = 2xb =gt b = 1/2):}`
`x^2 + x + (1/2)^2 - (1/2)^2 +2= (x + 1/2)^2 - 1/4 + 2 `
`x^2 + x + (1/2)^2 - (1/2)^2 +2 = (x + 1/2)^2 + 7/4`
You need to use the following substitution such that:
`x+1/2 = t => x = t- 1/2 => dx = dt`
Changing the variable to integrand yields:
`int (3x^2+1)/((x^2+x+2)^3)dx = int (3(t-1/2)^2+1)/((t^2+7/4)^3)dt`
You need to solve the integral `int (3(t-1/2)^2+1)/((t^2+7/4)^3)d` t using partil fraction decomposition such that:
`(3(t-1/2)^2+1)/((t^2+7/4)^3) = (At + B)/(t^2+7/4) + (Ct+D)/((t^2+7/4)^2) + (Et+F)/((t^2+7/4)^3)`
`3t^2 - 3t + 3/4 + 1 = (At + B)((t^2+7/4)^2) +...
(The entire section contains 356 words.)
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