Final answer:
This problem is a calculus question about integrating the function √(4 + x²). It involves using trigonometric substitution followed by integration by parts or a similar method and then back-substituting to x to find the indefinite integral with the constant of integration, C.
Step-by-step explanation:
The student is asking for help with a calculus problem specifically regarding the integration of the function f(x) = √(4 + x²). To calculate the integral of this function, you generally use trigonometric substitution. The expression x = 2tan(θ) is a suitable substitution because 1 + tan²(θ) = sec²(θ), and hence 4(1 + tan²(θ)) = 4sec²(θ). Thus, our integral becomes:
∫ 2sec^3(θ)dθ
To solve this integral, you might use integration by parts or look up a standard integral form. Don't forget to convert back to x using the original substitution after integrating with respect to theta (θ). The C at the end of the original integral represents the constant of integration, which is included in indefinite integrals.
Remember that the actual answer to this problem requires a set of steps including substitution, integration, and back-substitution, and is generally something tackled in high school or early college calculus courses.