Problem 46 - Essential Calculus

Question

Find the integral. \(\int \frac{2}{x \sqrt{1+4 x^{2}}} d x\)

Step-by-Step Solution

Answer

The integral of \( \int \frac{2}{x \sqrt{1+4x^{2}}} dx \) is \( \frac{\sqrt{1+4x^{2}}}{2} + C \)

Step 1: Identify a suitable substitution

The idea is to make the integral simpler by using substitution. A suitable substitution in this case is \(u = 1+4x^{2}\). This leads to \(du = 8x dx\). Dividing both sides by 8x gives \( \frac{1}{8x} du = dx\). Now, you can rewrite the integral equation using \(u\) and \(du\).

Step 2: Substitute and simplify

Plug \(u = 1 + 4x^{2}\) and \(dx = \frac{1}{8x} du\) into the integral to get \( \int \frac{2}{x \sqrt{u}} \cdot \frac{1}{8x} du\). This simplifies to \( \int \frac{1}{4u\sqrt{u}} du\).

Step 3: Integrate and substitute back

The remaining integral is easy to compute. The antiderivative of \(\frac{1}{4u\sqrt{u}}\) is \(\frac{\sqrt{u}}{2}\). Substituting \(u = 1 + 4x^{2}\) back, gives, \(\frac{\sqrt{1+4x^{2}}}{2}\). Dont forget to add the integration constant, \(C\).

Key Concepts

Substitution Method

The substitution method is a powerful tool in integral calculus to simplify complex integrals. It involves changing the variable of integration to make the integral easier to solve. This method is particularly useful when dealing with complicated functions that are difficult to integrate directly.

In our example, we have the integral \( \int \frac{2}{x \sqrt{1+4x^2}} \, dx \). To simplify this, we introduce a new variable \( u \) by setting \( u = 1 + 4x^2 \). This choice is strategic because it transforms the square root part of the function into a simpler form.

Once we have \( u \), we also need to express \( dx \) in terms of \( du \). Calculating the derivative, \( du = 8x \, dx \), and solving for \( dx \), we get: \( dx = \frac{1}{8x} \, du \).

This substitution allows us to rewrite the original integral into a form that is much easier to integrate.

Definite Integral

A definite integral is an integral calculated over a specific interval. In contrast to indefinite integrals, which include an arbitrary constant, definite integrals result in a numerical value representing the accumulated quantity over the specified range.

In situations like our exercise, however, though the substitution simplifies the process, the given integral appears to be indefinite due to the lack of specified limits. Nonetheless, it is essential to understand how definite integrals work:

• Definite integrals have limits of integration, say from \(a\) to \(b\).
• The outcome depends only on the values of the function between these limits.
• The integral sign includes these limits as lower and upper bounds, \( \int_{a}^{b} \).

Since our example doesn't provide bounds, we focus on understanding how to solve similar integrals by preparing them for potential definite integration.

Integration Techniques

Integration is the process of finding the integral of a function, which represents the area under its curve. As functions vary in complexity, different integration techniques are employed to tackle them. Some common techniques include:

• Substitution Method: Useful for expressions that include compositions of functions, like in our example.
• Integration by Parts: Suitable for products of functions, where one is easily integrable, and the other is differentiable.
• Partial Fractions: Applied typically to rational functions to break them into simpler fractions.

In our context, the substitution method was ideal, opening up a direct path to simplification and solution. By leveraging these techniques accordingly, one can solve a wide range of integrals efficiently.

Every technique has its purpose, and often, the trick is recognizing which one to apply based on the form of the integrand. Understanding the "why" and "how" behind each method is crucial, especially when tackling definite integrals, ensuring a smoother integration journey.