Techniques of Integration

Integration by parts is nothing but the product rule integrated. Suppose \(u,v\) are two differentiable functions. Then

\[(u(x)v(x))'=u(x)v'(x)+v(x)u'(x).\]

Moving terms and integrating both sides, we get

\[\int u(x)v'(x)\mathrm{d}x=uv-\int v(x)u'(x)\mathrm{d}x.\]

Any rational function \(\frac{P(x)}{Q(x)}\) has an antiderivative. The key to finding the antiderivative is to rewrite the rational function using polynomials of lower degree.

Theorem 1: Partial Fraction Decomposition (simplified version)
Suppose the degree of \(P\) is less than that of \(Q\) (if not, perform polynomial division). If \(Q\) has the form \(Q(x) = (x-a_1)(x-a_2)\cdots(x-a_n)\), a product of \(n\) distinct linear factors, then one can find constants \(A_1,\ldots,A_n\) such that \[\frac{P(x)}{Q(x)} = \frac{A_1}{(x-a_1)}+\frac{A_2}{(x-a_2)}+\cdots+\frac{A_n}{(x-a_n)}.\]

In the method of substitution, we substitute a function of \(x\) with the single variable \(u\), thus simplifying the expression. Inverse substitution is to substitute the single variable \(x\) with a function of a new variable. The most important inverse substitutions are the trigonometric substitutions.

Inverse sine substitution:

Let \(a>0\) and \(x \in (-a,a)\). The expression \(\sqrt{a^2-x^2}\) can be simplified by the substitution \(x=a\sin{\theta}\), yielding

\[\sqrt{a^2-x^2}=\sqrt{a^2(1-\sin^2\theta)}=a\cos\theta,\]

valid for \(\theta \in (-\pi/2,\pi/2)\).

Inverse secant substitution:

Let \(a>0\) and \(x \in (-\infty,-a) \cup (a,\infty)\). The expression \(\sqrt{x^2-a^2}\) can be simplified by the substitution \(x=a\sec\theta\), yielding

\[\sqrt{x^2-a^2} = a \sqrt{\sec^2\theta-1} = a \left| \tan\theta \right|. \]

It is valid for those \(\theta \in (-\pi/2,\pi/2)\) such that the expression under the square root is positive.

Inverse tangent substitution:

The expressions \(\sqrt{a^2+x^2}\) and \(\frac{1}{x^2+a^2}\) can be simplified by the substitution \(x=a\tan\theta\), yielding

\[\sqrt{a^2+x^2} = a\sec\theta, \qquad \frac{1}{x^2+a^2} = \frac{\cos^2\theta}{a^2}, \]

respectively. They are valid for \(a>0\) and \(\theta \in (-\pi/2,\pi/2)\).

The \(\tan(\theta/2)\)-substitution:

Any rational function of \(\sin(\theta)\) and \(\cos(\theta)\) can be turned into a rational function of \(x\), using the substitution \(x = \tan(\theta/2)\).

The Method of Undetermined Coefficients
Often one can deduce which class of functions the antiderivate belongs to from the integrand. For instance we know that the antiderivatives of polynomials are polynomials of one degree higher, and that the antiderivative of the exponential function is the exponential function itself. (In most cases where one can use this method, one can also use the other methods listed above, but this can sometimes be quicker.)

Integral tables
These are tables listing common indefinite integrals (see, e.g., Rottman).

Computer Algebra Systems
Maple, MATLAB (with Symbolic Math Toolbox) and WolframAlpha can all handle most indefinite integrals.

We have earlier defined the Riemann integral for bounded functions on bounded intervals. In practice, however, one needs to handle also the case when either the interval of integration, or the the integrand, is unbounded. This is done through so-called improper integrals, defined as limits of proper integrals.

Definition 1: Improper integrals of type I
If \(f\) is continuous on \([a,\infty)\), we define the improper integral of \(f\) over \([a,\infty)\) as

\[ \int_a^{\infty}f(x)dx=\lim_{R\to \infty}\int_a^R f(x)dx.\]

We say that the integral exists if the limit exists (i.e. as a finite value). Integrals over intervals of the form \((-\infty,b]\) are defined similarly.

We also define \[ \int_{-\infty}^{\infty}f(x)dx= \lim_{a \to -\infty} \int_a^0 f(x)dx + \lim_{b \to \infty} \int_0^b f(x)dx,\] so that the integral on the left is defined if both of the limits on the right exist.

Definition 2: Improper integrals of type II
If \(f\) is continuous on \((a,b]\) and unbounded near \(a\), we define the improper integral of \(f\) as

\[\int_a^b f(x)dx=\lim_{c\to a^+}\int_c^bf(x)dx.\]

The case where \(f\) is integrable over \([a,b)\) and unbounded near \(b\) is defined similarly, using a left-hand limit.

Some improper integrals are good to know by heart.

Theorem 2: p-integrals
If \(a > 0\), then

\[\begin{aligned} (a) \quad \int_a^{\infty}x^{-p}dx \quad \begin{cases} \mbox{converges to } \frac{a^{1-p}}{p-1} & \mbox{if } p>1, \\ \mbox{diverges to } \infty & \mbox{if } p \le 1. \end{cases}\\ (b) \quad \int_0^ax^{-p}dx \quad \begin{cases} \mbox{converges to } \frac{a^{1-p}}{p-1} & \mbox{if } p<1, \\ \mbox{diverges to } \infty & \mbox{if } p \ge 1. \end{cases}\\ \end{aligned}\]

Often we are not interested in the exact value of an improper integral, but only whether it converges or diverges. The following theorem is useful in this respect, and is often used in tandem with Theorem 2.

Theorem 3: A comparison theorem for integrals
Let \(-\infty \le a < b \le \infty\). Suppose \(f\) and \(g\) are continuous on the interval \((a,b)\) and satisfy \(0 \le f(x) \le g(x)\) there. If \(\int_a^b g(x) dx\) converges, then so does \(\int_a^b f(x) dx\), and \[ \int_a^b f(x) dx \le \int_a^b g(x) dx. \]

Sometimes it is not possible to find an antiderivative of the integrand in terms of familiar functions; for example, it is not possible to find a closed formula for \(\int \sin(x^2) dx\). But there are accurate and efficient numerical methods for approximating \(\int_a^b \sin(x^2)dx\), given real numbers \(a\) and \(b\).

The following three numerical integration methods all use the same strategy. Given a function \(f\) over an interval \([a,b]\), divide \([a,b]\) into \(n\) subintervals of equal length, say \(h=(b-a)/n\), and assume the value of \(f\) is known at the points \(x_0=a, x_1=a+h,\ldots, x_n=a+nh\). The idea is to, on each subinterval \([x_{i-1},x_i]\), approximate \(f\) by a simpler function, and integrate this simper function instead. What separates the methods is what these simpler functions are.

Definition 3: The Trapezoid Rule (linear approximation)
Let \(f\) be a bounded function over an interval \([a,b]\). We divide \([a,b]\) into \(n\) subintervals of equal length, say \(h=(b-a)/n\), and we assume the value of \(f\) is known at the points \(x_0=a, x_1=a+h,\ldots, x_n=a+nh\). Next we approximate \(f\) by linear interpolation between the points in the plane \((x_i,f(x_i))\). This creates trapezoids over each sub-interval with vertices \((x_{i-1},0)\), \((x_{i-1},f(x_{i-1}))\), \((x_i,f(x_i))\) and \((x_i,0)\). Recalling the formula for the area of a trapezoid, we get

\[\int_a^b f(x)\mathrm{d}x \simeq h\left(\frac{f(x_0)+f(x_1)}{2}+\ldots+\frac{f(x_{n-1})+f(x_n)}{2}\right).\]

Definition 4: The Midpoint Rule (approximation with a constant)
On each subinterval \([x_{i-1},x_i]\), approximate \(f\) by the constant \(f(m_i)\), where \(m_i = (x_{i-1}+x_i)/2\):

\[\int_a^b f(x)\mathrm{d}x \simeq h\left( f(m_1)+f(m_2)+\cdots+f(m_n) \right).\]
Definition 5: Simpson's Rule (approximation with a quadratic function)
This method requires that the number \(n\) of subintervals is even. With an even number of subintervals, we can join pairs of consecutive subintervals together to form \(n/2\) larger subintervals, each of length \(2h\) and with \(3\) known function values. On each of the larger subintervals we approximate \(f\) with the (unique) quadratic function that passes through these three points. This yields Simpson's rule:

\[\int_a^b f(x)\mathrm{d}x \simeq h\left( f(x_0)+4f(x_1)+2f(x_2)+4f(x_3)+\cdots+ 2f(x_{n-1})+f(x_{n}) \right),\] where the coefficient is \(4\) for odd indices, \(1\) at the endpoints and \(2\) for the rest of the even indices.

Finally, we stress that a numerical approximation is only as good as its error bounds. Theorem 4 and 5 in the book give precise error estimates for the above integration rules.