Problem 13
Question
Use partial fractions to find the integral. $$ \int \frac{x^{2}+3 x-4}{x^{3}-4 x^{2}+4 x} d x $$
Step-by-Step Solution
Verified Answer
The integral of \((x^{2}+3 x-4)/(x^{3}-4 x^{2}+4 x)\) dx is given by \(-2ln|x| + 6ln|x-2| +\frac{1}{x-2} + C\)
1Step 1: Factorize the denominator
Factorizing the denominator \(x^{3}-4 x^{2}+4 x\) gives \(x(x-2)^{2}\).
2Step 2: Decompose the integrand into partial fractions
Decompose the function \(\frac{x^{2}+3 x-4}{x(x-2)^{2}}\) into the form: \(\frac{A}{x}+ \frac{B}{(x-2)} + \frac{C}{(x-2)^{2}}\)Multiply both sides by the denominator on the right to equate the numerators which gives: \(x^{2}+3 x-4 = A(x-2)^{2} + Bx + Cx^{2}\). Substitute convenient values of x to find A, B, and C. For instance, for x=0, A=-2, for x=2, B=6, and by matching coefficients, C=-1.
3Step 3: Integrate using partial fractions
Perform the integration separately for each term, remembering to add the constant of integration C:\(\int \frac{-2}{x} dx + \int \frac{6}{x-2} dx - \int \frac{1}{(x-2)^2} dx = -2ln|x| + 6ln|x-2| +\frac{1}{x-2} + C\)
Key Concepts
Integral CalculusRational FunctionsFactorization
Integral Calculus
Integral calculus is a branch of mathematics that deals with integrating functions, which is essentially finding the integral or antiderivative of a function. An integral can be thought of as the reverse process of differentiation. The integral calculates the area under a curve or, more generally, provides a way to accumulate quantities, such as areas, volumes, and other concepts that arise from summing infinitesimal changes.
In the integral calculus, we often encounter complex expressions, requiring us to simplify or decompose them, just like in our example. This is where techniques like partial fraction decomposition are highly beneficial.
Some of the main steps in solving an integral include:
In the integral calculus, we often encounter complex expressions, requiring us to simplify or decompose them, just like in our example. This is where techniques like partial fraction decomposition are highly beneficial.
Some of the main steps in solving an integral include:
- Identifying the function to be integrated
- Determining if the function can be simplified
- Applying appropriate integration techniques
- Ensuring that the final result includes the constant of integration, which denotes the family of antiderivatives
Rational Functions
A rational function is a function that can be expressed as the quotient of two polynomials. For example, in our exercise, the function \(\frac{x^{2}+3x-4}{x^{3}-4x^{2}+4x}\)is a rational function because both the numerator and the denominator are polynomials.
Rational functions often exhibit interesting behavior, such as vertical and horizontal asymptotes, where their values approach infinity or become constant. These properties make them essential for modeling a wide range of phenomena in various scientific fields.
When handling rational functions in calculus, particularly in integration, simplifying them can make the process much more manageable. Partial fraction decomposition is one such technique that allows us to break down a complex rational function into simpler fractions, which can then be integrated more easily. This technique is often used in cases where the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Rational functions often exhibit interesting behavior, such as vertical and horizontal asymptotes, where their values approach infinity or become constant. These properties make them essential for modeling a wide range of phenomena in various scientific fields.
When handling rational functions in calculus, particularly in integration, simplifying them can make the process much more manageable. Partial fraction decomposition is one such technique that allows us to break down a complex rational function into simpler fractions, which can then be integrated more easily. This technique is often used in cases where the degree of the polynomial in the numerator is less than the degree of the polynomial in the denominator.
Factorization
Factorization involves breaking down an expression into products of simpler expressions. In mathematics, particularly algebra, factorizing polynomial expressions is a vital skill. For instance, factorizing the denominator of a rational function can simplify the expression, making calculations easier.
In our exercise, the denominator \(x^{3}-4x^{2}+4x\) is factorized into \(x(x-2)^{2}\), enabling us to apply partial fraction decomposition effectively. By expressing the denominator in terms of its factors, we facilitate the separation of the rational function into more manageable pieces.To factorize a polynomial, one often follows these steps:
In our exercise, the denominator \(x^{3}-4x^{2}+4x\) is factorized into \(x(x-2)^{2}\), enabling us to apply partial fraction decomposition effectively. By expressing the denominator in terms of its factors, we facilitate the separation of the rational function into more manageable pieces.To factorize a polynomial, one often follows these steps:
- Look for a common factor in all terms
- Apply techniques such as grouping, use of identities, or the quadratic formula
- Verify by expanding the factors to ensure they multiply back to the original expression
Other exercises in this chapter
Problem 13
In Exercises \(7-26,\) evaluate the limit, using \(L\) 'Hôpital's Rule if necessary. (In Exercise \(12, n\) is a positive integer.) \(\lim _{x \rightarrow 0} \f
View solution Problem 13
Use the Special Integration Formulas (Theorem 6.2) to find the integral. $$ \int \sqrt{25-4 x^{2}} d x $$
View solution Problem 13
Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int t \ln (t+1) d t $$
View solution Problem 14
Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{4}{\sqrt[4]{x}} d x $$
View solution