91Ó°ÊÓ

In integral calculus, a definite integral is an integral with upper and lower limits. It computes the net area under a curve within a specified range on the x-axis, often between two points.

The integral \(\int_{1}^{3} \frac{2 x+3}{x^{2}+3 x} d x\) guides us through such a calculation, bounded between 1 and 3.

Unlike indefinite integrals which result in a function plus a constant, definite integrals yield a specific value. To evaluate, apply the Fundamental Theorem of Calculus: \[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \] where F(x) is the antiderivative of f(x). This theorem makes definite integrals instrumental in practical applications across physics, engineering, and beyond.

Partial fraction decomposition is a method used to break down complex rational expressions into simpler fractions.
It's particularly useful when integrating fractions.

Consider the integral \(\int_{1}^{3} \frac{2 x+3}{x^{2}+3 x} d x\). Here, \(\frac{2x + 3}{x^2 + 3x}\) is decomposed into partial fractions:

\[ \frac{2x + 3}{x(x + 3)} = \frac{A}{x} + \frac{B}{x + 3} \]

By solving for constants A and B through the method of equating coefficients, we can simplify the integral further.
In our example, setting up and solving \((2x + 3 = A(x + 3) + Bx)\) leads to \(A = 1\) and \(B = 1\).These simpler fractions \(\frac{1}{x}\) and \(\frac{1}{x + 3}\) are then much easier to integrate.

Logarithmic integration, involving the natural logarithm function, often appears when integrating functions of the form \(\int \frac{1}{x} dx\).
For example, after decomposing \(\frac{2x + 3}{x^2 + 3x}\) via partial fractions, we separate it into integrals like \(\int_{1}^{3} \frac{1}{x} dx\).

Integrating \(\frac{1}{x}\) and \(\frac{1}{x+3}\) results in logarithmic functions:

\[ \int \frac{1}{x} \, dx = \ln|x| + C \] \[ \int \frac{1}{x + 3} \, dx = \ln|x + 3| + C \]

Using the Fundamental Theorem of Calculus, we evaluate these integrals from 1 to 3.
This process results in expressions involving \( \ln(3) \) and \( \ln(6) - \ln(4)\), combining to give the final result.

Simplifying integrals often involves rewriting the integral in a more manageable form.
For \(\int_{1}^{3} \frac{2 x+3}{x^{2}+3 x} d x\), factorize and decompose to break down into simpler integrals.

Following partial fraction decomposition, the integral becomes:
\[ \int_{1}^{3} \frac{1}{x} dx + \int_{1}^{3} \frac{1}{x + 3} dx \]

These integrals lead to the natural logarithms: \(\ln|x|\bigg|_{1}^{3}\) and \(\ln|x + 3|\bigg|_{1}^{3}\).
Use properties of logarithms to combine results:

• \[ \ln(3) \- \ln(1) = \ln(3) \] and \[ \ln(6) \- \ln(4)\]
• \[ \ln(3 \times \frac{6}{4}) = \ln(4.5)\]

This combination simplifies integration results, presenting the solution in the simplest form. Always aim to reduce to the most compact and understandable solution.