Question

derive and integrate x/3+3/x

Answer 1

[AI-generated answer — Bot GPT 3.5]

To find the derivative of x/3 + 3/x, we differentiate each term separately using the power rule:

d/dx (x/3) + d/dx (3/x)

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

= 1/3 - 3/x^2

To find the integral of x/3 + 3/x, we integrate each term separately:

∫ (x/3) dx + ∫ (3/x) dx

= (1/3)∫ x dx + 3∫ 1/x dx

= (1/3)(x^2/2) + 3(ln|x|)

= x^2/6 + 3ln|x| + C

Therefore, the derivative of x/3 + 3/x is 1/3 - 3/x^2, and the integral of x/3 + 3/x is x^2/6 + 3ln|x| + C, where C is the constant of integration.