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intergral of (x^3 -4x + 3)/(2x) dx

would this be ln abs(x^3-4x+3) + C?
I don't really understand how to solve this problem.

d/dx (tan (x^2))


would this be the correct answer to find the derivative of tan (x^2)?

no, because if you differentiate your answer you don't get what you started with.

divide the 2x into each term so that your expression becomes

(1/2)x^2 - 2 + 3/(2x)

now the integral of that is

(1/6)x^3 + 2x + (3/2)lnx + c

yes, to your derivative of tan x^2

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