`int 1/(x^2-7x+12)dx =?`

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You need to evaluate the given indefinite integral of rational function, hence, you need to check if you can factorize the denominator, such that:

`x^2 - 7x + 12 = (x - x_1)(x - x_2)`

You need to use quadratic formula to evaluate `x_(1,2)` , such that:

`x_(1,2) = (7 +- sqrt(49 - 48))/2 => x_(1,2) = (7 +- 1)/2`

`x_1 = 4 ; x_2 = 3`

`x^2 - 7x + 12 = (x - 4)(x - 3)`

Hence, you may re-write the integrand, such that:

`int 1/(x^2 - 7x + 12)dx = int 1/((x - 4)(x - 3))dx`

You need to use the partial fraction expansion, to split the integrand into two fractions in lowest terms, such that:

`1/((x - 4)(x - 3)) = a/(x - 4) + b/(x - 3)`

`1 = a(x - 3) + b(x - 4)`

`1 = ax - 3a + bx - 4b`

Grouping the terms that contain x yields:

`1 = x(a + b) - 3a - 4b`

Equating the coefficients of equal powers yields:

`{(a + b = 0),(-3a - 4b = 1):} => {(a = -b),(3b - 4b = 1):} => {(a = -b),(-b = 1):} => {(a = 1),(b = -1):}`

`1/((x - 4)(x - 3)) = 1/(x - 4) - 1/(x - 3)`

Integrating both sides, yields:

`int 1/((x - 4)(x - 3)) dx = int 1/(x - 4) dx - int 1/(x - 3) dx`

`int 1/((x - 4)(x - 3)) dx = ln|x - 4| - ln|x - 3| + c`

Using the logarithmic identities yields:

`int 1/((x - 4)(x - 3)) dx = ln|(x - 4)/(x - 3)| + c`

Hence, evaluating the given indefinite integral, using partial fraction expansion, yields `int 1/(x^2 - 7x + 12)dx = ln|(x - 4)/(x - 3)| + c` .

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