## To find the antiderivative of a function, you need to integrate it. In this case, the antiderivative of 1/(x^2-2x+2) can be found using a technique called partial fraction decomposition.

To solve this, you can start by factoring the denominator (x^2-2x+2). However, in this case, the denominator cannot be factored since it does not have any real roots. Therefore, you would need to use a different approach.

One way to approach this is by using a technique called completing the square. By completing the square, you can rewrite the denominator in a form that makes it easier to integrate. The completed square form of the denominator is (x-1)^2 + 1.

Now, you can rewrite 1/(x^2-2x+2) as 1/[(x-1)^2 + 1]. To integrate this, you can consider it as the derivative of the arctan function. The antiderivative of 1/[(x-1)^2 + 1] is arctan(x-1) + C, where C represents the constant of integration.

So, the correct antiderivative of 1/(x^2-2x+2) is arctan(x-1) + C.

Regarding your second question about finding the critical points of (x+2)^5*(x-3)^4:

To find the critical points, you need to find the values of x where the derivative is equal to zero or undefined.

You correctly found the derivative of the function, which is [5(x+2)^4*(x-3)^4] + [4(x+3)^3*(x-2)^5]. Now, set this derivative equal to zero and solve for x.

However, note that when solving for zeros, you need to be careful with any cancellations or simplification. Make sure to check your calculations and simplify the equation correctly. It's possible for some zeros to cancel out or not be considered critical points due to simplification errors.

Once you have the zeros, those can be considered as potential critical points. You can further analyze the behavior of the function around these points to determine if they are local maximum, local minimum, or neither, by performing the first and second derivative tests or by evaluating the function values at those points.

So, in summary, the antiderivative of 1/(x^2-2x+2) is arctan(x-1) + C, and three zeros can potentially be critical points based on your derivative calculation.