Problem 28 - Thomas Calculus

Question

Evaluate the integrals by using a substitution prior to integration by parts. \(\int \ln \left(x+x^{2}\right) d x\)

Step-by-Step Solution

Answer

Using substitution and integration by parts, the evaluated integral is \((x + x^2) \ln(x + x^2) - (x + x^2) + C\).

Step 1: Identify the substitution for simplification

To simplify the integrand, start by making a substitution. Let \( u = x + x^2 \), then differentiate with respect to \( x \): \( du = (1 + 2x) dx \). This substitution helps in transforming the integral into a simpler form.

Step 2: Express dx in terms of du

Solve the equation \( du = (1 + 2x) dx \) for \( dx \). From substitution \( x = \frac{-1 \pm \sqrt{1 + 4u}}{2} \). However, focus on simplifying using the substitution where \( dx = \frac{du}{1 + 2x} \).

Step 3: Rewrite the original integral using substitution

Substitute the values of \( u \) and \( dx \) into the integral. The integral becomes \( \int \ln(u) \frac{du}{1 + 2x} \). However, refine step by correctly identifying \( x \) in terms of \( u \) to consolidate simplification. Further transforming can yield easier handling.

Step 4: Integration by Parts Setup

Set up using integration by parts, for \( \int \ln(u) \cdot 1 \, du \). Use the formula: \[ \int v \, dw = vw - \int w \, dv \]. Set \( v = \ln(u) \) and \( dw = du \). Therefore, \( dv = \frac{1}{u} du \) and \( w = u \).

Step 5: Apply Integration by Parts

Apply the integration by parts formula:\[ \int \ln(u) \, du = u \ln(u) - \int 1 \, du \]Solve:\[ u \ln(u) - u + C \].

Step 6: Substitute back to the original variable

Replace \( u \) with \( x + x^2 \) to return to the original variable context of the integral:\[ (x + x^2) \ln(x + x^2) - (x + x^2) + C \].

Key Concepts

Substitution Method

The substitution method is a powerful tool in integration, enabling us to transform a complicated integral into a simpler form that is easier to evaluate. In essence, substitution involves changing variables to simplify the integral. For the given problem, the initial step starts with identifying a suitable substitution. We choose a substitution that can simplify the expression we are integrating. For this integral, let - \( u = x + x^2 \) This substitution targets the complex part within the logarithm in the integrand. By differentiating \( u \) with respect to \( x \), we find: - \( du = (1 + 2x) \, dx \)
Subsequently, we solve for \( dx \) in terms of \( du \) and the substitution variable. Here, \( dx = \frac{du}{1 + 2x} \). This expression allows us to replace \( dx \) in the integral, thus simplifying the process.

• Ensure when substituting back, find \( x \) in terms of \( u \) effectively to avoid computational errors.

Ultimately, an effective substitution rests on how well it reduces complexity and links the integrand’s features to a more manageable form.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation, commonly used when faced with an integral of the product of functions. It's particularly useful when direct integration is problematic or overly complex. With integration by parts, we use the formula: \[ \int v \, dw = vw - \int w \, dv \]
In the integral \( \int \ln(u) \cdot 1 \, du \), which emerges after substitution, we assign:

• \( v = \ln(u) \)
• \( dw = du \)

This choice leverages the simplicity of the derivative of the logarithm function. Therefore, we find:

• \( dv = \frac{1}{u} \, du \)
• \( w = u \)

Applying the integration by parts formula allows us to calculate: \[ u \ln(u) - \int u \, \frac{1}{u} \, du \]
What's notable here is the transformation to a new integral, \( \int 1 \, du \), which is straightforward to compute: simplifying to \( u \). Integration by parts highlights the strategic breakdown of integrals for ease in solving.

Logarithmic Integration

Logarithmic integration often appears when the integrand includes a natural logarithm, such as \( \ln(u) \). In these scenarios, methods like substitution and integration by parts are particularly relevant. The given exercise asks us to find the integral of \( \ln(x + x^2) \, dx \), initially transformed to \( \ln(u) \) after substitution.
Logarithms often simplify complex algebraic expressions, but integrating them directly requires a strategic approach like integration by parts. After substitution, the role of logarithmic integration in this context becomes prominent: managing the expression \( \ln(u) \) in a new integral form.

• Recognize how substitution not only transforms the log function but aligns well with integration by parts.
• Consider the expression within the logarithm to select the most effective integration technique.

By strategically leveraging these methods, logarithmic integration is effectively simplified to an elementary approach, allowing deeper exploration of log-involved integrals turning complex expressions manageable.