Problem 35 - Essential Calculus

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{2}^{4} \frac{2}{x \sqrt{x^{2}-4}} d x $$

Step-by-Step Solution

Answer

The integral is convergent and its exact value is 4

Step 1: Define the integral

Firstly, mark the integral to be evaluated, in this case, it is \( \int_{2}^{4} \frac{2}{x \sqrt{x^{2}-4}} dx \)

Step 2: Perform substitution

Substitute \( u = x^{2} - 4 \) therefore \( du = 2x.dx \) . Then, the integral becomes \( \int_{0}^{12} \frac{1}{\sqrt{u}} du \)

Step 3: Evaluate the integral

Now, we apply the power rule for integration which states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} \) for any real number n not equal to -1. It simplifies to \( 2\sqrt{u} \)

Step 4: Substitute back the variables

Substitute back \( u = x^{2} - 4 \) into the evaluated integral to get \( 2\sqrt{x^{2} - 4} \)

Step 5: Evaluate definite Integral

We now evaluate the definite integral at the upper limit and then subtract the evaluation at the lower limit to get \( 2\sqrt{4^2 - 4} - 2\sqrt{2^2 - 4} = 4 \)