Problem 53 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{d x}{2 \sqrt{x}+2 x}$$

Step-by-Step Solution

Answer

\(\ln|1+\sqrt{x}| + C\)

Step 1: Simplify the integrand

Start by factoring the denominator of the integrand. Notice that \(2\sqrt{x} + 2x\) can be factored as \(2(\sqrt{x} + x)\). Thus, the integral becomes: \[\int \frac{dx}{2(\sqrt{x} + x)} = \frac{1}{2}\int \frac{dx}{\sqrt{x} + x}\]

Step 2: Use a substitution method

To simplify the integral, use the substitution \(u = \sqrt{x}\). Then \(x = u^2\) and \(dx = 2u \, du\). Substitute these into the integral:\[\frac{1}{2} \int \frac{2u \, du}{u + u^2}\]Cancel the 2 in the integrand:\[\int \frac{u \, du}{u + u^2}\]This simplifies to:\[\int \frac{u \, du}{u(1+u)} = \int \frac{du}{1+u}\]

Step 3: Integrate the simplified expression

The resulting integral from the substitution is easier to evaluate:\[\int \frac{du}{1+u} = \ln|1+u| + C\]where \(C\) is the constant of integration.

Step 4: Back-substitute to original variable

Replace \(u\) with \(\sqrt{x}\) to get the result in terms of the original variable, \(x\):\[\ln|1+\sqrt{x}| + C\]This is the solution to the integral in terms of \(x\).

Key Concepts

Integration Techniques

Integration plays a crucial role in calculus, helping us find the area underneath curves and solve advanced problems. Among the various techniques used in integration, simplification of the integrand and substitution are often employed to tackle complex expressions.
Simplification involves rewriting the expression in an easier form to integrate. For instance, in our example problem, we first factor the denominator to make the integral more manageable. This is a typical first step when faced with a challenging integrand.
Different techniques suit different problems, but knowing how to simplify integrands and choose suitable methods is key. Mastering these skills will greatly improve your ability to solve integral calculus problems.

Substitution Method

The substitution method is a powerful tool in solving integrals, especially when faced with complex expressions. The core idea is to transform a difficult integral into a simpler one by changing variables.
In our exercise example, the substitution \(u = \sqrt{x}\) allows us to change the variable from \(x\) to \(u\). This substitution simplifies the integrand significantly. The derivatives are also transformed: \(dx = 2u \, du\). Thus, our integral becomes easier to handle:

• Transform the expression: \(x = u^2\), which changes \(dx = 2u \, du\).
• Modify the integral: Substitute all \(x\)-dependent parts with \(u\)-dependent parts.

Ultimately, this step changes a difficult integral into a simpler one, often involving basic functions that are easier to integrate.

Definite and Indefinite Integrals

Integrals in calculus are classified into definite and indefinite types. It's important to understand how they differ.
An indefinite integral finds a function, called the antiderivative, whose derivative is the original function. Typically represented with \(C\), indicating a constant of integration, these integrals do not have specified limits of integration. The solution reflects a general form.

• Example: \(\int f(x) \, dx = F(x) + C\)

Definite integrals, by contrast, calculate the net area between the function and the x-axis over a specified interval. They have upper and lower limits. Unlike indefinite integrals, definite integrals yield a numerical result.

• Example: \(\int_{a}^{b} f(x) \, dx\)

In our example problem, we evaluated an indefinite integral, which resulted in an expression with \(C\). Recognizing whether an integral is definite or indefinite is crucial as it influences how we approach the solution.