Question

Evaluate each integral. $$\int \frac{1}{x \sqrt{4 x^{2}-9}} d x$$

   Evaluate each integral.
$$\int \frac{1}{x \sqrt{4 x^{2}-9}} d x$$

Calculus

Dale Varberg, Edwin… 9th Edition

Chapter 6, Problem 71

Instant Answer

Solved by Expert Rukhmani Jain

06/13/2021

Step 1

This gives us: $$\int \frac{1}{x \sqrt{4 x^{2}-9}} d x = \int \frac{1}{x \sqrt{4(x^{2}-\frac{9}{4})}} d x$$ Show more…

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Evaluate each integral. $$\int \frac{1}{x \sqrt{4 x^{2}-9}} d x$$

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Key Concepts

Trigonometric Substitution

This technique is used to simplify integrals that involve square roots of quadratic expressions. By substituting a trigonometric expression for the variable, the square root simplifies using fundamental trigonometric identities, turning a complicated algebraic integral into one that is easier to evaluate.

Inverse Trigonometric Functions

After performing a trigonometric substitution, the antiderivative is typically expressed in terms of the substitution angle. Converting this back into the original variable often leads to an answer featuring an inverse trigonometric function, which expresses the original variable in a form corresponding to the trigonometric ratio used in the substitution.

Substitution Method

This is a fundamental integration technique that involves replacing a complicated expression within an integrand with a new variable. This simplification makes the integral easier to evaluate, and after integration, the substitution is reversed to express the result in terms of the original variable.

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