When dealing with integrals that contain square root terms, it is important to first try to simplify the expression. This can be done by factoring out any common factors or using trigonometric identities. If the integral cannot be simplified, then substitution or integration by parts may be necessary.
The general strategy for solving integrals with square root terms is to first try to simplify the expression, then use substitution or integration by parts if necessary. If the integral cannot be solved using these methods, then numerical approximation techniques may be used.
To solve integrals of the form ∫sqrt(a^2 + x^2) dx, we can use the trigonometric substitution x = a tan u. This will result in the integral becoming ∫a sec^2 u du, which can be easily solved using basic integration rules.
As an example, let's solve the integral ∫sqrt(4 + x^2) dx. We can use the trigonometric substitution x = 2 tan u. This will result in the integral becoming ∫2 sec^2 u du. Using the formula for integrating sec^2 u, we get 2 tan u + C. Substituting back in for u and simplifying, we get the final answer of 2x + 2√(4 + x^2) + C.
Integrals of the form ∫sqrt(2 + x^2) dx can be solved using the substitution x = √2 tan u. This will result in the integral becoming ∫√2 sec^2 u du. We can then use the formula for integrating sec^2 u to get √2 tan u + C. Substituting back in for u and simplifying, we get the final answer of √2x + 2√(2 + x^2) + C.