Final answer:
To evaluate the integral ∫(1/(x-6x²-6x+8))dx from 0, factor the denominator and use partial fractions to split the fraction into smaller fractions. Integrate each term separately and evaluate from 0 to find the final answer.
Step-by-step explanation:
To evaluate the integral ∫(1/(x-6x²-6x+8))dx from 0, we can first factor the denominator. The denominator can be factored as (x-2)(x-1) by using the quadratic formula. We can then rewrite the integral as ∫(1/((x-2)(x-1)))dx.
Next, we can use partial fractions to split the fraction into smaller fractions. We can write 1/((x-2)(x-1)) as A/(x-2) + B/(x-1), where A and B are constants.
Finally, we integrate each term separately. The integral of A/(x-2) is A ln|x-2| and the integral of B/(x-1) is B ln|x-1|. Evaluating these integrals from 0 gives us our final answer.