周次: 6 日期: March 30, 2022 节次: 1
In this lecture, we’ll learn a new technique for integration, called Integration by Parts. The second part of the lecture will be devoted to the partial fractions decomposition of rational functions. By using such technique, we are able to integrate all the rational functions!
§ 8.2 Integration by Parts
Integration by parts is a technique for simplifying integrals of the form
$$ \int f(x)g(x)dx. $$
It is useful when $f$ can be differentiated repeatedly and $g$ can be integrated repeatedly without difficulty. For example the integral $\int x e^x dx$ and $\int e^x \sin(x)d x$ can be such type.
Product Rule in Integral Form
Integration by Parts Formula
$$ \int u(x), v'(x) , dx = u(x), v(x) -\int u'(x) , v(x) , dx. $$
The mindset behind the formula is that to transfer a difficult integration to an easier one.
Sometimes it is easier to remember the formula if we write it in differential form.
$$ \int u dv = uv - \int v du. $$
Why it works?
The formula for integration by parts is coming from the Product Rule for derivatives
$$ \frac{d}{dx}\left(u(x) v(x)\right) = u'(x) v(x) + u(x) v'(x). $$
Example 1. Using Integration by Parts
Find
$$ \int x\cos(x) , dx. $$
Example 3. Integral of the Natural Logarithm
Find
$$ \int \ln(x) dx. $$
Example 4. Repeated Use of Integration by Parts
Find
$$ \int x^2 e^x dx $$
Example 5. Solving for the Unknown Integral (An Circling-Back Integration)
Find
$$ \int e^x \cos(x) dx. $$
Evaluating Definite Integrals by Parts
Combining the Fundamental Theorem of Calculus with the Product Rule, we find the following formula for definite integrals
$$ u(x) v(x) \bigg]_a^b = \int_a^b u'(x) v(x) dx + \int_a^b u(x) v'(x) dx. $$
Therefore we can find the following integration by parts formula for definite integrals.
Integration by Parts Formula for Definite Integrals
$$ \int_a^b u(x) v'(x) dx= u(x) v(x) \bigg]_a^b - \int_a^b u'(x) v(x) dx. $$
Example 6. Finding Area
Find the area of the region bounded by the curve $y=x e^{-x}$ and the $x$-axis from $x=0$ to $x=4.$
Example 7. Repeating Multiple Times. Ft. The Tabular Integration Method
Find
$$ \int x^3 \sin(x) dx. $$
Example 9. A Reduction Formula (Iterative Formulas)
Find the integral
$$ I_n = \int \cos^n(x) dx $$
Example 10. Using the Reduction Formula
Find
$$ \int \cos^3(x) dx. $$
§ 8.3 Integration of Rational Functions by Partial Fractions
This section shows how to express a rational function as a sum of simpler fractions, called the method of partial fractions, which are easily integrated. For example
$$ \dfrac{5x-3}{x^2-2x -3} = \dfrac{2}{x+1} + \dfrac{3}{x-3}. $$
By doing so, an anti-derivative can be found easily.
$$ \int \dfrac{5x-3}{x^2-2x-3} dx = 2\ln|x+1| + 3\ln|x-3| + C. $$
**Note Any rational function can be integrated by using this method!**
The General Description of the Method
The method is not a guess work. Its efficiency can be proved rigorously. To show case by the previous example, for
$$ \dfrac{5x-3}{x^2-2x-3} $$
we first remark on the degrees: the degree of $5x-3$ is lower than the degree of $x^2-2x-3$, which is good! Then, factoring $x^2-2x-3$ gives two factors $x+1$ and $x-3$. Therefore, we can pretend the rational function is a linear combination of partial fractions
$$ \dfrac{5x-3}{x^2-2x-3} = \dfrac{A}{x+1}+\dfrac{B}{x-3}. $$
We call $A$ and $B$ undetermined coefficients.
To find $A$ and $B$, simply clear the above fractions, and compare the coefficients of like powers of $x$ on the two-sides.
$$ A+B = 5,\quad -3A + B = -3. $$
Solving these equations simultaneously gives $A=2$ and $B=3.$
Method of Partial Fractions ( for $f(x)/g(x)$ Proper)
- Factorize the denominator $g(x)$, obtain all its linear and quadratic factors, as well as their powers. In general, the denominator has the factorization looks like
$$ \dfrac{f(x)}{g(x)} = \dfrac{f(x)}{(x-r_1)^{m_1}\cdots (x-r_k)^{m_k} (x^2+p_1 x + q_1)^{n_1}\cdots(x^2+p_l x + q_l)^{n_l}} $$
where all the linear and quadratic factors are distinct. And $x^2+p_i x +q_i$ should not be factorized anymore ( called irreducible ).
**Note The feasibility of this factorization is guaranteed by the Fundamental Theorem of Algebra.**
- For each linear factor $(x-r)^m$ of $g(x)$, the fraction $f(x)/g(x)$ should include the following partial fractions, with undetermined coefficients $A_k$.
$$ \dfrac{A_1}{(x-r)} + \dfrac{A_2}{(x-r)^2} + \cdots \dfrac{A_m}{(x-r)^m} $$
- For each quadratic factor $(x^2+px+q)^n$ of $g(x)$, the obtained partial fractions should include
$$ \dfrac{B_1 x + C_1}{x^2+px+q} + \dfrac{B_2 x + C_2}{(x^2+px+q)^2} +\cdots + \dfrac{B_n x + C_n}{(x^2+px+q)^n} $$
- Set the original fraction $f(x)/g(x)$ equal to the sum of all these partial fractions. Clear the resulting equation of fractions and arrange the terms in decreasing powers of $x$.
- Equate the coefficients of corresponding powers of $x$ and solve the resulting equations for the undetermined coefficients.
Example 1. Distinct Linear Factors
Evaluate
$$ \int \dfrac{x^2+4x+1}{(x-1)(x+1)(x+3)}dx $$
using partial fractions.
( Solution: $\frac34 \ln|x-1|+\frac12\ln |x+1| -\frac14 \ln |x+3| + C$ .)
Example 2. A Repeated Linear Factor
Evaluate
$$ \int \dfrac{6x+7}{(x+2)^2}dx $$
( Solution: $6\ln|x+2| + \frac{5}{x+2} + C$ )
Example 3. Integrating an Improper Fraction
Evaluate
$$ \int \dfrac{2x^3-4x^2-x-3}{x^2-2x-3}dx $$
( Solution: $x^2 + 2\ln |x+1| + 3\ln |x-3| + C$ )
Example 4. Integrating with an Irreducible Quadratic Factor
Evaluate
$$ \int \dfrac{-2x+4}{(x^2+1)(x-1)^2}dx $$
(Solution: $\ln(x^2+1) + \arctan x - 2\ln|x-1| - \frac1{x-1} + C$ )
Example 5. A Repeated Irreducible Quadratic Factor
Evaluate
$$ \int \dfrac{dx}{x(x^2+1)^2}. $$
( Solution: $\ln \dfrac{|x|}{\sqrt{x^2+1}} + \dfrac1{2(x^2+1)} + C.$ )
The Heaviside “Cover-up” Method for Linear Factors
For proper fraction $f(x)/g(x)$, if $g(x) = (x-r_1) (x-r_2) \cdots (x-r_n)$ is a product of $n$ distinct linear factors, each raised to the first power, there is a quick way to find the undetermined coefficients.
Learn Heaviside “Cover-up” Method by Example
Find $A$, $B$ and $C$ in the partial-fraction expansion
$$ \dfrac{x^2+1}{(x-1)(x-2)(x-3)}= \dfrac{A}{x-1} + \dfrac{B}{x-2} + \dfrac{C}{x-3}. $$
Example 7. Integrating with the Heaviside Method
Evaluate
$$ \int \dfrac{x+4}{x^3+3x^2-10 x}dx. $$
( Solution : $-\frac25 \ln |x| + \frac37 \ln |x-2| - \frac1{35} \ln |x+5| + C$. )
Other ways to Determine the Coefficients
Example 8. Using Differentiation
Find $A$, $B$ and $C$ in the equation
$$ \frac{x-1}{(x+1)^3}= \dfrac{A}{x+1} + \dfrac{B}{(x+1)^2} + \dfrac{C}{(x+1)^3}. $$
( Solution : $A=0$, $B=1$ , $C=-2$ )
Example 9. Assigning Numerical Values to $x$ (Equivalent to “Cover-up” Method)
Find $A$, $B$ and $C$ in
$$ \dfrac{x^2+1}{(x-1)(x-2)(x-3)} = \dfrac{A}{x-1} + \dfrac{B}{x-2} + \dfrac{C}{x-3} $$
( Solution : $A=1$, $B=-5$, $C=5$ )