Problem 5 - Active Calculus

Question

The form of the partial fraction decomposition of a rational function is given below. \(\frac{25 x-10 x^{2}-45}{(x-5)\left(x^{2}+9\right)}=\frac{A}{x-5}+\frac{B x+C}{x^{2}+9}\) \(A=\) ______________ \(B=\) _____________ \(C=\) ______________ Now evaluate the indefinite integral. \(\int \frac{25 x-10 x^{2}-45}{(x-5)\left(x^{2}+9\right)} d x=\) ____________________

Step-by-Step Solution

Answer

A = -5, B = -5, C = 0\(\int \frac{25x - 10x^2 - 45}{(x -5)(x^2 + 9)} dx = -5 \ln|x - 5| - \frac{5}{2} \ln|x^2 + 9| + C\)

Step 1 - Set up the partial fractions

Given: \[ \frac{25x - 10x^2 - 45}{(x - 5)(x^2 + 9)} = \frac{A}{x - 5} + \frac{Bx + C}{x^2 + 9} \] Multiply both sides by \((x - 5)(x^2 + 9)\) to clear the denominators: \[ 25x - 10x^2 - 45 = A(x^2 + 9) + (Bx + C)(x - 5) \]

Step 2 - Expand and simplify

Expand the right-hand side: \[ 25x - 10x^2 - 45 = A(x^2 + 9) + Bx(x - 5) + C(x - 5) \] \[ 25x - 10x^2 - 45 = Ax^2 + 9A + Bx^2 - 5Bx + Cx - 5C \]

Step 3 - Combine like terms

Combine like terms on the right-hand side of the equation: \[ 25x - 10x^2 - 45 = (A + B)x^2 + (C - 5B)x + 9A - 5C \]

Step 4 - Set up the system of equations

Match the coefficients of corresponding powers of \(x\) to form equations: \[ -10 = A + B \] \[ 25 = C - 5B \] \[ -45 = 9A - 5C \]

Step 5 - Solve the system of equations

Solve for \(A\), \(B\), and \(C\): \[ -10 = A + B \rightarrow (1) \] \[ 25 = C - 5B \rightarrow (2) \] \[ -45 = 9A - 5C \rightarrow (3) \]From (1): \[ A = -10 - B \] Substitute \(A\) into (3): \[ -45 = 9(-10 - B) - 5C \] \[ -45 = -90 - 9B - 5C \] \[ 45 = -9B - 5C \rightarrow (3') \] Use (2) for \(C\): \[ C = 25 + 5B \] Substitute \(C\) in (3'): \[ 45 = -9B - 5(25 + 5B) \] \[ 45 = -9B - 125 - 25B \] \[ 170 = -34B \] \[ B = -5 \] Substitute \(B = -5\) in \(A = -10 - B\): \[ A = -10 + 5 = -5 \] Substitute \(B = -5\) into \(C = 25 + 5B\): \[ C = 25 - 25 = 0 \]

Step 6 - Indefinite integral

Now that \(A\), \(B\), and \(C\) are found, rewrite the partial fraction decomposition: \[ \int \frac{25x - 10x^2 - 45}{(x - 5)(x^2 + 9)} dx = \int \left( \frac{-5}{x - 5} + \frac{-5x}{x^2 + 9} \right) dx \] Evaluate each integral separately: \[ \int \frac{-5}{x - 5} dx = -5 \ln|x - 5| \] \[ \int \frac{-5x}{x^2 + 9} dx \] Use substitution \(u = x^2 + 9\), \(du = 2x dx\), then \(x dx = \frac{1}{2} du\): \[ \int \frac{-5x}{x^2 + 9} dx = -\frac{5}{2} \int \frac{du}{u} = -\frac{5}{2} \ln|u| = -\frac{5}{2} \ln|x^2 + 9| \] Thus, combining both integrals: \[ -5 \ln|x - 5| - \frac{5}{2} \ln|x^2 + 9| + C \]

Key Concepts

Indefinite integral

An indefinite integral represents the family of functions that differentiates to a given function. When you encounter an indefinite integral, it means you are finding a function whose derivative is the integrand (the function inside the integral). For instance, if \(\frac{d}{dx} F(x) = f(x)\), then \(\frac{d}{dx} \int f(x) \, dx = f(x)\). The general form of an indefinite integral includes a constant of integration, 'C', because the derivative of a constant is zero.

Rational functions

Rational functions are fractions where the numerator and denominator are both polynomials. A common task with rational functions is to decompose them into simpler fractions, called partial fractions, to make integration easier. For example, the rational function \(\frac{25x - 10x^2 - 45}{(x - 5)(x^2 + 9)}\) was decomposed into the partial fractions: \(\frac{-5}{x - 5} + \frac{-5x}{x^2 + 9}\). This simplifies the integration process.

System of equations

Solving a system of equations is often necessary to find constants in partial fraction decompositions. A system of equations is a set of two or more equations with the same variables. You solve the system by finding the values of the variables that satisfy all equations simultaneously. In the provided solution, setting up the system from coefficients led to three equations:\([-10 = A + B]\),\([25 = C - 5B]\), and\([-45 = 9A - 5C]\), which were solved to find \(A = -5\), \(B = -5\), and \(C = 0\).

Integration by substitution

Integration by substitution is a method used to simplify the integration process. It involves substituting a part of the integrand with a new variable to make the integral easier. For instance, in the integral\( \int \frac{-5x}{x^2 + 9} \, dx\), the substitution \( u = x^2 + 9 \) and \( du = 2x \, dx \) was used. This transforms the integral into \( \int \frac{-5}{2} \frac{du}{u} \), which is easier to evaluate. This technique helps break down more complex integrals into basic forms that are straightforward to solve.