91Ó°ÊÓ

Integration by parts is a powerful technique used to integrate products of functions, based on the formula: \[ \int u \, dv = uv - \int v \, du \]. This method is particularly useful when you have an integral of the form \( \int u \, dv \), where choosing parts of the original integral can simplify the solution.In our exercise, after choosing a suitable \( u \)-substitution \( u = \sqrt{x} \), we encountered the product \( u e^u \). Here, integration by parts helps us tackle this:

• Set \( u = u \) and hence \( dv = e^u \, du \).
• Differentiate to find \( du = du \) and integrate for \( v = e^u \).

The formula becomes \[ 2(ue^u - \int e^u \, du) \], where after evaluating \( \int e^u \, du \), we get the integral solved. Integration by parts seamlessly breaks down complex expressions into simpler parts, making it an essential tool in calculus.

Understanding definite and indefinite integrals is key to mastering calculus. An indefinite integral represents a family of functions and includes an arbitrary constant \( C \) because the integral of a function can differ by a constant when derived. The process is expressed as \[ \int f(x) \, dx = F(x) + C \].In contrast, a definite integral computes the net area under a curve within specific bounds, typically \( a \) to \( b \). It's calculated as:\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \].In our exercise, we calculated the indefinite integral \( \int e^{\sqrt{x}} \, dx \). The result, \( 2e^{\sqrt{x}}(\sqrt{x} - 1) + C \), represents a family of curves that only differ by their vertical shift, represented by the constant \( C \). Mastering these concepts will enlighten many real-life applications, from finding displacement in physics to determining cost functions in economics.

Change of variables, also known as \( u \)-substitution, is an essential method that helps to simplify integrals. It involves substituting part of the integral with a new variable \( u \) to transform the integral into a more manageable form. In our task, change of variables plays a pivotal role:

• Select a substitution, such as \( u = \sqrt{x} = x^{1/2} \) in our exercise, reducing complexity by transforming \( x \) in terms of \( u \).
• Express \( dx \) in this new variable, where \( dx = 2u \, du \) after differentiation.

Rewriting the original integral \( \int e^{\sqrt{x}} \, dx \) to \( \int e^u \cdot 2u \, du \) demonstrates how integrating with respect to \( u \) simplifies the problem. By changing variables, we align the integral with known methods or formulas, often like those found in integral tables, making integration accessible even for seemingly complex expressions.