Question

Decide whether you can find the integral \(\int \frac{2 d x}{\sqrt{x^{2}+4}}\) using the formulas and techniques you have studied so far. Explain your reasoning.

Short answer

Yes, it's possible to calculate the integration of the provided function. The integral is \(2ln|x+\sqrt{x^{2}+4}|+C\).

Step-by-step solution

Identify the Type of Integral

The given function is \(\int \frac{2 dx}{\sqrt{x^{2}+4}}\). It has the general form of a rational function where the numerator is a constant, and the denominator, under square root, is a sum of a squared variable (x) and a constant (4). This resembles a standard integral derived from the hyperbolic sine function.

Check for Standard Integral Form

Verifying whether it's in the standard integral form, for any function of type \(\int \frac{dx}{\sqrt{x^{2}+a^{2}}}\), its integral is \(ln|x+\sqrt{x^{2}+a^{2}}|+C\). Here, \(a = 2\). So the function can be integrated using this formula.

Compute the Integral

Applying the formula, for the integral \(\int \frac{2 dx}{\sqrt{x^{2}+4}}\), we should multiply '2' with the derived formula. Thus, the integral is \(2ln|x+\sqrt{x^{2}+4}|+C\). This expression is the indefinite integral.