91Ó°ÊÓ

Integration is a fundamental concept in calculus, and there are various techniques to solve different types of integrals.
When faced with a complex integral like \( \int \frac{d x}{x \sqrt{1+4 x^{2}}} \), choosing the right technique can prove crucial. Various techniques include but are not limited to:

• Basic substitution
• Integration by parts
• Partial fractions
• Trigonometric substitution

For the integral in question, trigonometric substitution is ideal. This technique is typically used when integrals involve expressions of the form \( \sqrt{a^2 + u^2} \) or similar.
This approach transforms the integral into a form that is easier to evaluate by utilizing trigonometric identities, making it a powerful tool for tackling integrals containing complex algebraic expressions.

The substitution method is a common technique in integration that simplifies complex expressions by substituting a new variable.
In our example, we use trigonometric substitution, which involves substituting a trigonometric function for a variable to take advantage of trigonometric identities. Given our integral \( \int \frac{d x}{x \sqrt{1+4 x^{2}}} \), we choose a substitution that transforms the square root expression into a simpler form:

• Set \( x = \frac{1}{2}\tan\theta \)
• Differential becomes \( dx = \frac{1}{2}\sec^2\theta \, d\theta \)
• The square root simplifies to \( \sec\theta \)

This substitution leverages the identity \( \sqrt{1 + \tan^2\theta} = \sec\theta \).
Substituting these values into the integral simplifies the problem drastically, allowing us to rewrite the complex integral into one with an easier solution.

An antiderivative is a fundamental concept in calculus representing a function whose derivative is a given function. In other words, finding an antiderivative is pivotal in solving integrals, as it identifies the original function before differentiation.
When we consider \( \int \frac{d x}{x \sqrt{1+4 x^{2}}} \), and apply trigonometric substitution, it simplifies to finding the antiderivative of \( \cot\theta \). The antiderivative of \( \cot\theta \) is \( \ln|\sin\theta| \), which is straightforward when we apply:

• Integrate \( \int \cot\theta \, d\theta = \ln|\sin\theta| + C \)
• Then back-substitute \( \tan\theta = 2x \)
• Convert \( \sin\theta \) back in terms of \( x \) for the final expression

Finally, using these steps, we return to the variable \( x \) to find the antiderivative as \(-\frac{1}{2}\ln(1 + 4x^2) + C \).
Antiderivatives showcase the powerful interplay between differentiation and integration, enabling us to revert from a derivative to find the original function.