Problem 40 - Essential Calculus

Question

Decide whether you can find the integral \(\int \frac{2 d x}{\sqrt{x^{2}+4}}\) using the formulas and techniques you have studied so far. Explain your reasoning.

Step-by-Step Solution

Answer

Yes, it's possible to calculate the integration of the provided function. The integral is \(2ln|x+\sqrt{x^{2}+4}|+C\).

Step 1: Identify the Type of Integral

The given function is \(\int \frac{2 dx}{\sqrt{x^{2}+4}}\). It has the general form of a rational function where the numerator is a constant, and the denominator, under square root, is a sum of a squared variable (x) and a constant (4). This resembles a standard integral derived from the hyperbolic sine function.

Step 2: Check for Standard Integral Form

Verifying whether it's in the standard integral form, for any function of type \(\int \frac{dx}{\sqrt{x^{2}+a^{2}}}\), its integral is \(ln|x+\sqrt{x^{2}+a^{2}}|+C\). Here, \(a = 2\). So the function can be integrated using this formula.

Step 3: Compute the Integral

Applying the formula, for the integral \(\int \frac{2 dx}{\sqrt{x^{2}+4}}\), we should multiply '2' with the derived formula. Thus, the integral is \(2ln|x+\sqrt{x^{2}+4}|+C\). This expression is the indefinite integral.

Key Concepts

Definite Integrals

Definite integrals are all about finding the area under a curve. When we compute a definite integral of a function over a specific interval, we are essentially calculating the total accumulation of the quantity represented by the function. Think of it like measuring the total distance traveled by a car over a given time period. In mathematical terms, a definite integral from point \(a\) to \(b\) is denoted as \( \int_{a}^{b} f(x) \, dx \).

• The limits of integration, \(a\) and \(b\), give us the interval on the x-axis we are focusing on.
• The output of a definite integral is a number, representing the area between the curve of the function and the x-axis over this interval.

Definite integrals are extremely useful in real-world scenarios where you want to find total changes or accumulations like areas, volumes, and more.

Indefinite Integrals

Indefinite integrals, on the other hand, are a bit like finding the recipe for the function's area by itself. Unlike definite integrals that give you a specific number, indefinite integrals produce a family of functions. They are represented without limits as \( \int f(x) \, dx \).

• Indefinite integrals are concerned with finding the antiderivative, a function whose derivative is the original function you started with.
• When computing an indefinite integral, we often include a '+C' at the end to represent that there are infinitely many possible antiderivatives, offset by a constant.

Indefinite integrals tell us the general pattern of a function, helping us understand how it might accumulate over any interval. For example, in the given exercise, we learned that \(\int \frac{2 dx}{\sqrt{x^{2}+4}} = 2 \ln|x+\sqrt{x^2+4}| + C\), which represents an indefinite integral with many possible solutions depending on the constant \(C\).

Integral Formulas

Integral formulas are essential tools in calculus, much like a handy cheat sheet. They help simplify the process of finding integrals by providing standard solutions for frequently encountered functions. The step-by-step solution to the exercise used such a formula, making the problem easier to solve.

• For example, one common integral formula is \(\int \frac{dx}{\sqrt{x^2+a^2}} = \ln|x+\sqrt{x^2+a^2}| + C\). This formula was directly used to solve the given exercise by recognizing the integral's form.
• Using these formulas, you can quickly integrate without performing lengthy algebra each time, as long as the problem can be rewritten to match a known form.

Familiarity with integral formulas is a huge advantage, speeding up the integration process and providing insights into the patterns of calculus.