Evaluate the integrals in Exercises 53–76.
57. ∫dx/(x√(25x²-2))

Guida verificata passo dopo passo

Identify the integral to solve: \(\int \frac{dx}{x \sqrt{25x^{2} - 2}}\).

Recognize that the integrand contains a square root of a quadratic expression of the form \(a^{2}x^{2} - b^{2}\), which suggests using a trigonometric substitution to simplify the square root.

Set up the substitution by letting \(x = \frac{b}{a} \sec(\theta)\), where \(a = 5\) and \(b = \sqrt{2}\). So, let \(x = \frac{\sqrt{2}}{5} \sec(\theta)\).

Compute \(dx\) in terms of \(d\theta\) by differentiating the substitution: \(dx = \frac{\sqrt{2}}{5} \sec(\theta) \tan(\theta) d\theta\).

Rewrite the integral entirely in terms of \(\theta\), simplify the expression, and then integrate using trigonometric identities.

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