# How do you evaluate #int dx/(x^2-2x+2)# from #[0, 2]#?

The denominator cannot be factored, so complete the square and see what you can do in the way of u-substitution.

This is a standard integral.

Evaluate using the 2nd fundamental theorem of calculus.

Hopefully this helps!

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To evaluate the integral ∫dx/(x^2 - 2x + 2) from 0 to 2, you can use the method of partial fraction decomposition followed by integration. First, factor the denominator x^2 - 2x + 2 as (x - 1)^2 + 1. Then, express the integrand as a sum of two fractions with unknown constants A and B:

1 / ((x - 1)^2 + 1) = A / (x - 1) + B / ((x - 1)^2 + 1)

Solve for A and B by finding a common denominator and equating coefficients of like terms:

A((x - 1)^2 + 1) + B(x - 1) = 1

Once you have found the values of A and B, you can integrate each term separately. The integral of A / (x - 1) can be evaluated using the natural logarithm function, and the integral of B / ((x - 1)^2 + 1) can be evaluated using the arctangent function.

Finally, evaluate the definite integral from 0 to 2 using the fundamental theorem of calculus. Substitute the upper and lower limits into the antiderivatives of the fractions and subtract the result of the lower limit from the result of the upper limit.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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