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derivative am i right or wrong

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what is the antiderivative of

does antiderivative mean find the derivative of the function or integrate it.

is the answer ln(x^2-2x+2) or
is-2(x^2-2x+2)^-2. These are part of the multiple choices.

antiderivative is the integral.

so would the answer be ln(x^2-2x+2).

also i need help with finding critical points of (x+2)^5*(x-3)^4

i got the derivative of that which is

[5(x+2)^4*(x-3)^4] + [4(x+3)^3*(x+2)^5] and i set it to zero and got a total of three zeros. i don't need to solve for them but are three zeros correct.

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