91Ó°ÊÓ

Integral calculus is a fundamental concept used to determine the accumulation of quantities and the area under a curve. In this context, the goal is to evaluate an integral using partial fraction decomposition—a powerful technique that simplifies complex rational functions into simpler ones.Partial fraction decomposition involves breaking down a complicated rational expression into a sum of simpler fractions. This makes it easier to integrate each term. For the given integral, \[ \int \frac{4 x^{2}+3 x+1}{(x+1)^{2}(x-1)} \, dx, \] we need to identify the denominator components and use them for decomposition. These components include linear factors like \((x-1)\) and repeated linear factors such as \((x+1)^2\).Breaking down polynomials this way allows integral calculus to transform difficult integrals into a manageable series of tasks. Ultimately, the solution involves integrating each simpler fraction individually, transforming an initial complex problem into multiple simpler ones.

A system of equations is pivotal in solving for the unknown constants in partial fraction decomposition. After representing the original expression in terms of partial fractions, we derive a set of equations reflecting the coefficients of each corresponding power of \(x\).In our example, when we clear the denominator and expand the equation,\[ 4x^2 + 3x + 1 = A(x+1)(x-1) + B(x-1) + C(x+1)^2, \] we obtain coefficients:

• \(A + C = 4\)
• \(B + 2C = 3\)
• \(-A - B + C = 1\)

Solving this system of equations provides the values of \(A\), \(B\), and \(C\) as 5, -1, and -1, respectively.Utilizing algebra, this step ensures that every term corresponds correctly to the formula, equipping us with the necessary components for the subsequent integration task.

Integration techniques are essential for solving the decomposed partial fractions. After determining the partial fractions, we integrate each term separately. This process is simplified due to the well-understood formulas for integrating basic rational expressions.For instance,

• For \(\int \frac{5}{x+1} \, dx\), the result is \(5 \ln|x+1| + C_1\).
• Integrating \(\int \frac{-1}{(x+1)^{2}} \, dx\) yields \(\frac{1}{x+1} + C_2\).
• Likewise, \(\int \frac{-1}{x-1} \, dx\) becomes \(-\ln|x-1| + C_3\).

Once these integrals are calculated separately, they are combined into a single solution:\[5 \ln|x+1| + \frac{1}{x+1} - \ln|x-1| + C,\]where \(C = C_1 + C_2 + C_3\) represents the overall constant of integration.These integration techniques illustrate the power of partial fraction decomposition and make complex integrals much more approachable.