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

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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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주요 개념

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Integration involving radicals

Integrals containing expressions with square roots, such as √(25x² - 2), often require substitution or trigonometric methods to simplify the integrand. Recognizing the form under the radical helps determine the appropriate technique.

Trigonometric substitution

Trigonometric substitution replaces algebraic expressions involving square roots with trigonometric functions to simplify integration. For expressions like √(a²x² - b), substitutions such as x = (b/a) sec θ are commonly used to transform the integral into a trigonometric form.

Integration of rational functions

When the integrand is a rational function or can be manipulated into one, techniques like substitution or partial fractions help evaluate the integral. Understanding how to rewrite the integrand is key to applying these methods effectively.

Evaluate the integrals in Exercises 53–76.57. ∫dx/(x√(25x²-2))