Final answer:
The integral ∫ dx / x2√x2 + 4 can be evaluated using trigonometric substitution by setting x = 2tan(θ), simplifying the integral, and then integrating with respect to θ. The final step includes back-substituting to express the answer in terms of x and adding the constant of integration.
Step-by-step explanation:
To evaluate the integral ∫ dx / x2√x2 + 4, we can apply trigonometric substitution. Let's set x = 2tan(θ), which means that dx = 2sec2(θ)dθ. By doing so, our expression under the square root becomes √(4tan2(θ) + 4) = 2√(1 + tan2(θ)) = 2√(sec2(θ)) = 2sec(θ).
Now substituting into our integral, we have ∫ (2sec2(θ)dθ) / (4tan2(θ)2sec(θ)), which simplifies to ∫ 1 / (2tan2(θ))dθ. This can be further simplified using the identity tan2(θ) = sec2(θ) - 1, and our integral is now ∫ 1 / (2(sec2(θ) - 1))dθ, or ∫ 1 / (2sec2(θ) - 2)dθ.
Integrating with respect to θ, we find the antiderivative in terms of θ, and then we back-substitute using our original substitution x = 2tan(θ) to return to the original variable x. The final step is to add the constant of integration, C, to get the complete solution.