91Ó°ÊÓ

Integration by parts is a technique used to solve integrals that are products of two functions. This method is derived from the product rule of differentiation and can be represented by the formula:

• \( \int u \, dv = uv - \int v \, du \)

In our problem, this technique helps solve the integral \( \int \frac{e^x}{x} \, dx \). Here, we choose:

• \( u = \ln x \)
• \( dv = e^x \, dx \)

From these choices, we derive:

• \( du = \frac{1}{x} \, dx \)
• \( v = e^x \)

Substituting these into the integration by parts formula, we evaluate the full expression step-by-step. This process helps us express the integral in terms of the non-elementary function \( F(x) \), resulting in:

• \( e^2 \ln 2 - F(2) + F(1) \)

Using integration by parts frequently helps in unraveling complex integrals, especially when one component becomes simpler upon differentiation.

The substitution method, or change of variables, is a fundamental technique in integration. It simplifies integrals by transforming them into a familiar or simpler form. This often involves substituting part of the integral with a new variable, \( u \), and then finding \( du \) in terms of \( dx \).In solving \( \int \frac{1}{\ln x} \, dx \), we choose substitution to tackle the complexity of the natural logarithm in the denominator. Letting \( u = \ln x \) and hence \( du = \frac{1}{x} \, dx \), the integral is transformed:

• \( \,\int_2^3 \frac{1}{\ln x} \, dx = \int_{\ln 2}^{\ln 3} \frac{1}{u} e^u \, du \)

After substitution, the problem resembles a structure more closely aligned with the function \( F(x) = \int_0^x e^{\varepsilon^t} \, dt \). Successfully applying the substitution method allows us to evaluate this integral in terms of \( F \), appearing as:

• \( F(\ln 3) - F(\ln 2) \)

The Fundamental Theorem of Calculus (FTC) unifies the process of differentiation with integration, providing a powerful connection between these operations. It states that if a function is integrable on an interval, its integral's derivative is the original function. This theorem is critical when evaluating non-elementary integrals with continuous function representations.In our example, we deal with \( F(x) = \int_0^x e^{\varepsilon^t} \, dt \), which is not an elementary function but can still be analyzed thanks to FTC. It assures us that \( F(x) \) is continuous and differentiable, although it may not be expressed in simple elementary terms. FTC thus helps express and evaluate integrals like:

• \( \int_1^2 \frac{e^x}{x} \, dx \)
• \( \int_2^3 \frac{1}{\ln x} \, dx \)

By integrating in terms of \( F \), whose existence is validated by the FTC, these integrals become more manageable.

Exponential functions, characterized by expressions like \( e^x \), are core components in many calculus problems. They have unique properties, such as their derivative and integral being proportional to \( e^x \), making them both simple yet powerful when it comes to integration.In the given integrals, \( e^x \) plays a significant role:

• In \( \int \frac{e^x}{x} \, dx \), the presence of \( e^x \) dictated the need for integration by parts.
• In \( \int \frac{1}{\ln x} \, dx \), the exponential transformation post-substitution aligned the integral with non-elementary function \( F(x) \).

Such properties and behaviors are what make exponential functions pivotal. They simplify certain steps but also give rise to non-elementary expressions when integrated in contexts involving non-polynomial-like terms. Consequently, understanding exponential functions helps in formulating expressions and finding solutions in terms like \( F(x) \).