5 More Integration
5.1 More Formulas
\(\int x^rdx=\frac{x^{r+1}}{r+1}+C\)
\(\int \sin x dx=-\cos x +C\)
\(\int \cos x dx=\sin x +C\)
\(\int \sec^2 x dx =\tan x+C\)
\(\int \frac{1}{1+x^2}dx=\tan^{-1}x+C\)
\(\int e^xdx=e^x +C\)
\(\int \frac{1}{x}dx=\ln|x| +C\)
5.2 Subsitution Method
Let \(y\) be a composition function of \(F\) with \(g\): \[y=F(g(x)).\] Suppose that the function \(g\) is differentiable on an open interval \(I\) and that the function \(F\) is differentiable on an open interval containing the image of \(I\) under \(g\). By the chain rule, the composition function \(F\circ g\) is differentiable on \(I\): \[\frac{d}{dx}F(g(x))=\frac{dy}{dx}=\frac{dy}{d g(x)}\frac{d g(x)}{dx}=F'(g(x))g'(x),\]
Since the integration is the reverse process of differentiation, we have \[\int F'(g(x))\cdot g'(x)dx=\int f(g(x))\cdot g'(x)dx=F(g(x))+C,\] where \(F'=f\).
Change of variable method Putting \(u=g(x)\) and using \(du=g'(x)dx\), we get \[\int f(g(x))g'(x)dx=\int f(u)du=F(u)+C.\]
Remarks The notations \(du\) and \(dx\) are called differentials. They are related by the fact that if \(\Delta x\) is small, then \(\frac{\Delta u}{\Delta x}\) is approximately equal to \(g'(x)\), that is \[\Delta u=g'(x)\Delta x.\] In the limiting situation, we have \(du=g'(x)dx\).
Example Find \(\int(x^2+1)^2 2x dx, \int\frac{\ln x}{x}dx, \int x^2e^{x^3}dx, \int\sin(2x-3)dx\)
Substitution method for definite integrals \[\int_a^bf(g(x))g'(x)dx=\int_{g(a)}^{g(b)}f(u)du\] where \(g\) is a continuous function on \([a,b]\) and \(f\) is a function defined and continuous on an open interval \(I\) containing the image of \([a,b]\) under \(g\).
Example Evaluate \(\int_0^4x\sqrt{x^2+9}dx, \int_0^1(x+1)e^{x^2+2x}dx, \int_0^{\frac{\pi}{2}}\sin x\cos xdx\)
Example Find the area of the region that lies between the \(x\)-axis and the graph of \(y=xe^{-x^2}\) for \(-1\le x\le 2\).
5.3 Integration of Rational Functions
\[\int \frac{A}{ax+b}dx\\=A\int\frac{1}{ax+b}dx\\=A\cdot\frac{1}{a}\cdot\ln|ax+b|+C\] We consider three cases when integrate \[\frac{Ax+B}{ax^2+bx+c}\]
Case 1 \(b^2-4ac>0\)
The denominator can be factorized as \(a(x-x_1)(x-x_2)\). Moreover, there exists contants \(\alpha, \beta\) such that \[\frac{Ax+B}{ax^2+bx+c}=\frac{\alpha}{x-x_1}+\frac{\beta}{x-x_2}\] The above process is called partial-fraction decomposition of the rational function.
Example Find \(\int\frac{x+1}{x^2-2x-3}dx\).
Case 2 \(b^2-4ac=0\).
The denominator can be factorized as \(a(x-x_1)^2\). Moreover, there exists contants \(\alpha, \beta\) such that \[\frac{Ax+B}{ax^2+bx+c}=\frac{\alpha}{x-x_1}+\frac{\beta}{(x-x_1)^2}.\]
Example Find \(\int\frac{2x+3}{x^2-2x+1}dx\)
Case 3 \(b^2-4ac<0\)
The denominator can be written as \(a((x+s)^2+t^2)\)
Subcase 3a \(A=0\) \[\int\frac{Ax+B}{ax^2+bx+c}dx=\frac{B}{at}\tan^{-1}\frac{x+s}{t} +C\]
Subcase 3b \(Ax+B=k(2ax+b)\) \[\int\frac{Ax+B}{ax^2+bx+c}dx=\int\frac{k(2ax+b)}{ax^2+bx+c}\] which can be integrated using substitution \(u=ax^2+bx+c\).
Case 3 in general We rewrite the numerator as a sum of two terms: the first one is a multiple of the derivative of the denominator and the second one is a constant.
Example Find \(\int\frac{2x+3}{x^2+4x+13}dx\)
5.4 Integration by Parts
The technique in integration that corresponds to the product rule in differentiation is called integration by parts.
Let \(f\) and \(g\) be functions that are differentiable on an open interval \((a, b)\). By the product rule, we have \[\frac{d}{dx}\left( f(x)g(x) \right)=f'(x)g(x)+f(x)g'(x), ~~ a<x<b,\] which, written in terms of integration, becomes \[\int f'(x)g(x)dx + \int f(x)g'(x)dx =f(x)g(x), ~~ a<x<b. \]
Or we have \[\int f(x)g'(x)dx = f(x)g(x) - \int f'(x)g(x)dx, ~~ a<x<b,\] in which \(\int f'(x)g(x)dx\) is easier to find than \(\int f(x)g'(x)dx\).
Example Find \(\int xe^xdx, \int x\cos xdx, \int \ln x dx\).
5.5 More Applications of Definite Integrals
Consumers’ and Producer’s surplus
Probability
Definition A variable whose values depend on the outcome of a random process is called a random variable.
Example
Suppose a die is rolled and \(X_1\) is the number that turns up. Discrete random variable
The life (in months) of a certain computer part. Continuous random variable
Definition Let \(X\) be a discrete random variable with values in \(\{x_1,x_2,\ldots,x_n\}\). A probability function of \(X\) is a function \(f\) with domain \(\{x_1,x_2,\ldots,x_n\}\) such that
\(0\le f(x_i)\) for all \(i=1,\ldots,n\);
\(f(x_1)+\cdots+f(x_n)=1\).
Probabilities of Events for Discrete Random Variables Suppose that \(X\) is a discrete random variable with values in the set \(\{x_1,\ldots,x_n\}\) and that \(f\) is a probability function of \(X\).
An event for \(X\) is a subset of \(\{x_1,\ldots,x_n\}\).
The probability of an event \(E\), denoted by \(P(E)\), is the number given by \[P(E)=\sum_{x_i\in E}f(x_i).\]
Definition Let \(X\) be a continuous random variable in \([a,\infty)\). A probability function of \(X\) is a function \(f\) with domain \([a,\infty)\) such that
\(0\le f(x_i)\) for all \(x\in[a,\infty)\);
\(\int_a^\infty f(x)dx=1\).
Remark \(\int_a^\infty f(x)dx\) is called an improper integral and is defined by \[\int_a^\infty f(x)dx=\underset{R\rightarrow\infty}{\lim}\int_0^Rf(x)dx,\] provided that the limit exists.
Example Find \(\int_1^\infty \frac{1}{x^2}dx\).
Probabilities of Events for Continuous Random Variables Suppose that \(X\) is a continuous random variable with values in the interval \((a,\infty]\) and that \(f\) is a probability function of \(X\).
An event for \(X\) is usually an interval contained in \([a,\infty)\); such events are represented by \((\alpha\le X\le \beta)\).
The probability of an event \((\alpha\le X\le\beta)\), denoted by \(P(\alpha\le X\le\beta)\), is the number given by \[P(\alpha\le X\le\beta)=\int_\alpha^\beta f(x)dx.\]
Remark It does not matter whether we include the endpoints \(\alpha,\beta\). For a continuous random variable \(X\), the probability that \(X\) equals a specific value is \(0\).