91Ó°ÊÓ

When evaluating integrals, an important concept is the integration of sums. The given integral can be split into two separate integrals because of the linearity of integrals. We leverage the property: \( \int \left( f(x) + g(x) \right) \, dx = \int f(x) \, dx + \int g(x) \, dx \) This makes it easier to handle each part individually. For our example, we start with \( \int_{1}^{e} \left( x + \frac{1}{x} \right) \ dx \). We can rewrite this as: \( \int_{1}^{e} x \, dx + \int_{1}^{e} \frac{1}{x} \, dx \) You can see this produces two straightforward integrals, which we can evaluate separately. This simplification often makes complex integrals more manageable and approachable.

Definite integrals are used to find the exact area under a curve between two points. They have fixed limits and provide a precise value instead of a general form. The general setup for a definite integral looks like this: \( \int_{a}^{b} f(x) \, dx \) Here, 1 and \(e \) are the lower and upper limits, respectively. It's essential to evaluate the antiderivative at these limits and then subtract. In the solution above, for both terms, it followed the pattern: \( F(b) - F(a) \) Where \( F(x) \) is the antiderivative of \( f(x) \), evaluated from \( x = a \) to \( x = b \). This process gives an exact numerical result for the integral.

One unique feature of integrals is how they handle specific functions, like the natural logarithm. The integral of \( \frac{1}{x} \) plays a notable role, resulting in the natural logarithm function. The formula we use is: \( \int \frac{1}{x} \, dx = \ln|x| + C \) This is because \( \frac{d}{dx}(\ln|x|) = \frac{1}{x} \). In our example, the integral term \( \int_{1}^{e} \frac{1}{x} \, dx \) evaluates to: \( [\ln|x|]_{1}^{e} = \ln(e) - \ln(1) \) Because \( \ln(e) = 1 \) and \( \ln(1) = 0, \) this simplifies to 1. Therefore, the understanding of how to integrate the natural logarithm gives us important results, particularly when evaluating definite integrals.