What is the result of the integral ∫ (x^2 + 1)^(1/2) dx?

To evaluate the integral ∫ (x^2 + 1)^(1/2) dx, one effective method is to use a trigonometric substitution. By substituting ( x = \tan(\theta) ), we can rewrite the integral in a way that allows for easier integration. This substitution gives us ( dx = \sec^2(\theta) d\theta ) and transforms ( (x^2 + 1)^{1/2} ) into ( \sec(\theta) ).

Thus, the integral becomes:

[

\int (x^2 + 1)^{1/2} dx = \int \sec(\theta) \sec^2(\theta) d\theta = \int \sec^3(\theta) d\theta

]

The integral of ( \sec^3(\theta) ) can be computed using the known formula:

[

\int \sec^3(\theta) d\theta = \frac{1}{3} \sec(\theta) \tan(\theta) + C

]

Substituting back our original variables (since ( \sec(\theta) = \sqrt{x^2 + 1} ) and