Questions: ∫ from 2 / 5 to 4 / 5 of (√(25x^2-4))/x dx

Transcript text: $\int_{2 / 5}^{4 / 5} \frac{\sqrt{25 x^{2}-4}}{x} d x$

Solution

Solution Steps

To solve the given integral, we can use a substitution method to simplify the expression under the square root. A trigonometric substitution might be appropriate here due to the form of the expression $ \sqrt{25x^2 - 4} $. After substitution, the integral can be evaluated, and then the result can be transformed back to the original variable.

Step 1: Define the Integral

We need to evaluate the definite integral

\[ I = \int_{2/5}^{4/5} \frac{\sqrt{25x^2 - 4}}{x} \, dx. \]

Step 2: Evaluate the Integral

After applying the appropriate substitution and evaluating the integral, we find that

\[ I \approx 1.3697. \]