91Ó°ÊÓ

Definite integrals are a fundamental aspect of integral calculus. They are used to find the net area between a function's graph and the x-axis within specific limits. The notation \( \int_{a}^{b} f(x) \, dx \) represents the definite integral of \( f(x) \) from \( a \) to \( b \), where \( a \) and \( b \) are the lower and upper limits, respectively.
When you evaluate a definite integral, the result is a number that represents the size of this area, taking into account parts above and below the x-axis. This can mean portions below the x-axis subtract from the total area, indicating the 'net' value. This concept applies directly to our exercise, where we integrate \( \int_{1}^{4} \frac{e^{\sqrt{x}}}{2 \sqrt{x}} \, dx \). By calculating this integral over the interval from 1 to 4, we find the accumulated value of the function, considering both top and bottom sections.
To solve a definite integral:

• Find the antiderivative of the function, unrelated to the limits initially.
• Evaluate this antiderivative at the upper limit.
• Evaluate it again at the lower limit and subtract this from the result at the upper limit.

By doing so, you'll get the net change of the function over the specified interval.

The substitution method, also known as \( u \)-substitution, is a technique for simplifying integrals. It's especially useful when dealing with composite functions, where one function is nested inside another. Think of it as a way to "undo" the chain rule of differentiation.
To use substitution:

• Identify the inner function, typically seen within another function or operation.
• Set this inner function equal to \( u \). For example, in \( \int \frac{e^{\sqrt{x}}}{2 \sqrt{x}} \, dx \), letting \( u = \sqrt{x} \) simplifies things.
• Differentiate the new \( u \) to express \( dx \) in terms of \( du \). In our case, \( dx = 2u \, du \).
• Rewrite the integral in terms of \( u \) and simplify before integrating.

This method not only simplifies calculations but may also make integration possible when it seems complex at first glance. Substitution realigns the limits based on the new variable \( u \), allowing us to perform integration more straightforwardly.

Antiderivatives are the heart of integrating functions. They refer to the reverse process of differentiation. In simple terms, if differentiating a function \( F(x) \) gives us \( f(x) \), then \( F(x) \) is the antiderivative of \( f(x) \). For definite integrals, finding the antiderivative is a key step before evaluating at the limits.
In our covered example, after using substitution, the integral simplifies to \( \int_{1}^{2} e^u \, du \). The antiderivative of \( e^u \) is simply \( e^u \) itself. Recognizing antiderivatives quickly is crucial for solving integrals efficiently.
When evaluating, you substitute back the variable values and compute the difference, as seen when calculating \( e^2 - e^1 \), ultimately resulting in \( e^2 - e \). Integrating through antiderivatives, especially after substitution, simplifies complex integrals in calculus considerably.