91Ó°ÊÓ

Definite integrals represent the signed area under the curve of a function in a specific interval—between an upper and a lower limit. These limits are denoted at the top and bottom of the integral sign. For example, in our original exercise, we want to evaluate \[ \int_{0}^{1} \frac{x+4}{x^{2}+3 x+2} \, dx \].
This symbol indicates that we want to find the net area from 0 to 1 under the curve described by the fractional function.

• Start by performing any necessary algebraic simplifications on the function, such as factoring or partial fraction decomposition.
• Integrals can often produce functions with logarithms or polynomials, depending on the integrand's structure.

The main part involves substituting the upper and lower limits into the antiderivative. This results in calculating the difference between the evaluation at these points. In simpler terms, if \[ F(x) \] is an antiderivative of \[ f(x) \], then the definite integral from \[ a \] to \[ b \] is given by:\[ F(b) - F(a) \].
This method gives us the total area we seek.

Partial fraction decomposition is a crucial step in integrals involving complex rational functions. By breaking a fraction down into simpler components, we make the integral easier to solve. In our example, we start with:\[ \frac{x+4}{(x+1)(x+2)} \].
The goal is to express this as:\[ \frac{A}{x+1} + \frac{B}{x+2} \].
Follow these actions:

• Clear the denominators by multiplying through by the factored denominator expression.
• Compare coefficients of corresponding powers of \[ x \] to solve for constants \[ A \] and \[ B \].

In this exercise, algebraic manipulation gives us \[ A = 3 \] and \[ B = -2 \], simplifying the integral's components. This simplification is instrumental in making the integration more straightforward, allowing us to integrate each term independently.

Factoring quadratic expressions is an essential skill, especially when dealing with rational functions in integrals. A quadratic expression is typically of the form \[ ax^2 + bx + c \].
For the given \[ x^2 + 3x + 2 \], the process involves finding two numbers that multiply to \[ c \] (in this case, 2) and add to \[ b \] (3).

• The quadratic \[ x^2 + 3x + 2 \] factors into \[ (x+1)(x+2) \].
• This step is pivotal as it sets up for partial fraction decomposition, making rational functions manageable.

Factoring simplifies the original problem considerably. It lets us identify simpler parts—like \[ (x+1) \] and \[ (x+2) \]—which can be addressed individually in further computations, such as finding constants in partial fractions.