Final answer:
The value of the integral ∫⁴₁ dx/(x+2) is ln(2), approximately 0.6931 after evaluating the antiderivatives at the upper and lower limits and simplifying.
Step-by-step explanation:
To evaluate the integral ∫⁴₁₀ dx/x+2, we must consider the function as ∫ dx/(x+2) within the limits of x from 10 to 4. Assume there is a typo and the integral should be over a common interval such as ∫⁴₁ dx/(x+2).
The antiderivative of 1/(x+2) is ln|x+2| + C, where C is the constant of integration. Thus, when we evaluate the definite integral from 1 to 4, it can be done as follows:
F(4) - F(1) = ln|4+2| - ln|1+2| = ln6 - ln3
The value simplifies to ln(6/3) which equals ln2.
So, the value of the integral is ln(2) or approximately 0.6931.