Other Integration Techniques and Symbolic Tools, calculus lesson

By now you have substitution, integration by parts, trigonometric methods and partial fractions. Real problems still produce integrals that fit none of them neatly, and some have no formula at all. This page collects the remaining tools, from smart guessing to tables and computer algebra, plus the one habit that keeps all of them honest. Differentiate your answer.

Key terms

Undetermined coefficients in antiderivatives

Definition When the form of an antiderivative is known except for some constants, you can write that form with unknown coefficients, differentiate it, and match coefficients with the integrand. The standard case is this. If $p$ is a polynomial of degree $n$ and $a \neq 0$ , then

\int p(x) e^{ax} \, dx = q(x) e^{ax} + C,

where $q$ is the unique polynomial of degree $n$ with $q'(x) + a\,q(x) = p(x)$ for all $x$ . Similarly, for real $a$ and $b$ not both zero,

\int e^{ax} \cos(bx) \, dx = e^{ax} \left(A \cos(bx) + B \sin(bx)\right) + C

for suitable constants $A$ and $B$ , and the same form works for $e^{ax} \sin(bx)$ .

In plain words If you know what shape the answer must have, write that shape with letters for the unknown numbers and let differentiation tell you the numbers. It replaces repeated integration by parts with a little algebra.

Example To integrate $x^2 e^x$ , try $(Ax^2 + Bx + C_0)e^x$ . Its derivative is

\left(Ax^2 + (2A + B)x + (B + C_0)\right)e^x.

Matching with $x^2 e^x$ gives $A = 1$ , $2A + B = 0$ and $B + C_0 = 0$ , so $B = -2$ and $C_0 = 2$ . Hence

\int x^2 e^x \, dx = (x^2 - 2x + 2)e^x + C.

Differential binomials

Definition A differential binomial is an integral of the form

\int x^m (a + b x^n)^p \, dx,

where $m$ , $n$ and $p$ are rational numbers, $n \neq 0$ , and $a$ and $b$ are nonzero real constants, considered on an interval where the integrand is defined. A classical theorem says the antiderivative is an elementary function if and only if at least one of the three numbers

p, \qquad \frac{m + 1}{n}, \qquad \frac{m + 1}{n} + p

is an integer. In the three cases the substitutions $x = t^N$ (with $N$ a common denominator of $m$ and $n$ ), $a + b x^n = t^s$ , and $a x^{-n} + b = t^s$ (with $s$ the denominator of $p$ ) turn the integrand into a rational function of $t$ .

In plain words For integrals made of a power of $x$ times a power of a two term sum, a quick integer test tells you whether a formula exists at all, and if so, which substitution finds it.

Example In the integral of $x^3 \sqrt{1 + x^2}$ you have $m = 3$ , $n = 2$ , $p = \frac{1}{2}$ , and $\frac{m + 1}{n} = 2$ is an integer. The theorem’s substitution is $1 + x^2 = t^2$ , and here the plain $u = 1 + x^2$ is even quicker. Put $u = 1 + x^2$ , so $x \, dx = \frac{du}{2}$ and $x^2 = u - 1$ . Then

\int x^3 \sqrt{1 + x^2} \, dx = \frac{1}{2} \int (u - 1) u^{1/2} \, du = \frac{1}{5}(1 + x^2)^{5/2} - \frac{1}{3}(1 + x^2)^{3/2} + C.

By contrast, the integrand $\sqrt{1 + x^3}$ has $m = 0$ , $n = 3$ , $p = \frac{1}{2}$ , and the three numbers $\frac{1}{2}$ , $\frac{1}{3}$ and $\frac{5}{6}$ are not integers. It has no elementary antiderivative.

Reduction strategies

Definition A reduction formula expresses an integral $I_n$ that depends on an integer $n$ in terms of integrals of the same kind with smaller index, such as $I_{n-1}$ or $I_{n-2}$ . It is usually found by integration by parts. For example, for every integer $n \ge 2$ ,

\int \sin^n x \, dx = -\frac{1}{n} \sin^{n-1} x \cos x + \frac{n - 1}{n} \int \sin^{n-2} x \, dx.

Repeating the formula lowers the index until only the integral of $1$ or of $\sin x$ remains, and those are easy.

In plain words Instead of attacking a high power head on, find a rule that knocks the power down by one or two, then use it again and again.

Example On $[0, \frac{\pi}{2}]$ the boundary term $\sin^{n-1} x \cos x$ is $0$ at both ends, so for $n \ge 2$

\int_0^{\pi/2} \sin^n x \, dx = \frac{n-1}{n} \int_0^{\pi/2} \sin^{n-2} x \, dx.

Therefore

\int_0^{\pi/2} \sin^4 x \, dx = \frac{3}{4} \cdot \frac{1}{2} \cdot \int_0^{\pi/2} dx = \frac{3}{8} \cdot \frac{\pi}{2} = \frac{3\pi}{16}.

Tables of integrals

Definition A table of integrals is a list of antiderivative formulas, each valid under stated conditions on its parameters and on the variable, such as

\int \frac{du}{u^2 + a^2} = \frac{1}{a} \arctan \frac{u}{a} + C, \qquad a \neq 0.

To use an entry, bring your integral into the exact form of the entry by algebra or by a substitution $u = g(x)$ , include the factor $du = g'(x)\,dx$ , and check that the stated conditions hold.

In plain words A table is a catalogue of finished answers. Most of the work is reshaping your integral until it matches one of them exactly.

Example To integrate $1/(4x^2 + 9)$ , put $u = 2x$ , so $du = 2 \, dx$ and $4x^2 + 9 = u^2 + 3^2$ . The entry with $a = 3$ gives

\int \frac{dx}{4x^2 + 9} = \frac{1}{2} \int \frac{du}{u^2 + 9} = \frac{1}{6} \arctan \frac{2x}{3} + C.

Computer algebra

Definition A computer algebra system is software that manipulates expressions exactly, with symbols and rational numbers rather than decimal approximations. For an integrand built from elementary functions it can search for an elementary antiderivative, and for large classes of integrands it can decide that none exists. Its output is an antiderivative on some set of $x$ values that it chooses, often under unstated assumptions about parameters (for example that a parameter is positive) and with branches of logarithms or roots that may not match your interval.

In plain words These programs are fast and usually right, but they answer the question they think you asked. Read the answer with the same care you would give a table entry.

Example Asked to integrate $\sin x \cos x$ , one system may print $\frac{1}{2}\sin^2 x$ , another $-\frac{1}{2}\cos^2 x$ , and another $-\frac{1}{4}\cos 2x$ . All three are correct, because

\frac{1}{2}\sin^2 x - \left(-\frac{1}{2}\cos^2 x\right) = \frac{1}{2}, \qquad \frac{1}{2}\sin^2 x - \left(-\frac{1}{4}\cos 2x\right) = \frac{1}{4}

are constants. Likewise an answer of $\ln x$ for the integral of $1/x$ is fine for $x > 0$ but needs to be read as $\ln \lvert x \rvert$ on $(-\infty, 0)$ .

Non-elementary antiderivatives

Definition The elementary functions are those built from constants and $x$ using finitely many arithmetic operations, roots, exponentials, logarithms, trigonometric functions, inverse trigonometric functions and compositions. Some elementary functions have no elementary antiderivative. Examples are

e^{-x^2}, \qquad \frac{\sin x}{x}, \qquad \frac{1}{\ln x}, \qquad \sqrt{1 + x^3}.

They still have antiderivatives on any interval where they are continuous. By the fundamental theorem of calculus, if $f$ is continuous on an interval $I$ and $c \in I$ , then the function

F(x) = \int_c^x f(t) \, dt, \qquad x \in I,

satisfies $F'(x) = f(x)$ for every $x \in I$ .

In plain words “No formula” does not mean “no antiderivative”. The antiderivative exists, it just cannot be written with the usual functions, so you name it as an integral and compute its values numerically.

Example Define

F(x) = \int_0^x e^{-t^2} \, dt.

Then $F'(x) = e^{-x^2}$ for every real $x$ , and numerical integration gives $F(1) \approx 0.7468$ . A multiple of this function is so useful in probability that it has its own name, the error function

\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} F(x).

Differentiation check

Definition A function $F$ is an antiderivative of $f$ on an interval $I$ if $F'(x) = f(x)$ for every $x \in I$ . If $F$ and $G$ are both antiderivatives of $f$ on the same interval, then $F - G$ is constant there.

In plain words Whatever method or tool produced an answer, differentiate it. If you get the integrand back on the whole interval you care about, the answer is right, and it can differ from anyone else’s correct answer only by a constant.

Example Suppose you claim that for $x > 0$

\int \ln x \, dx = x \ln x - x + C.

Differentiating gives

\frac{d}{dx}(x \ln x - x) = \ln x + x \cdot \frac{1}{x} - 1 = \ln x,

so the claim holds on $(0, \infty)$ .

Check yourself

Use undetermined coefficients to find $\int x e^{2x} \, dx$ .

Show the answer

Try $(Ax + B)e^{2x}$ . Its derivative is $(2Ax + A + 2B)e^{2x}$ , so $2A = 1$ and $A + 2B = 0$ . That gives $A = \frac{1}{2}$ , $B = -\frac{1}{4}$ , and the answer $\left(\frac{x}{2} - \frac{1}{4}\right)e^{2x} + C$ .

Does $\int x \sqrt{1 + x^3} \, dx$ have an elementary antiderivative?

Show the answer

No. Here $m = 1$ , $n = 3$ , $p = \frac{1}{2}$ , and the numbers $p = \frac{1}{2}$ , $\frac{m+1}{n} = \frac{2}{3}$ and $\frac{2}{3} + \frac{1}{2} = \frac{7}{6}$ are not integers.

A tool returns $-\frac{1}{2}\cos^2 x$ for $\int \sin x \cos x \, dx$ . Check it by differentiation.

Show the answer

The chain rule gives

\frac{d}{dx}\left(-\frac{1}{2}\cos^2 x\right) = -\cos x \cdot (-\sin x) = \sin x \cos x,

so it is a correct antiderivative on all of $\mathbb{R}$ .

Other Integration Techniques and Symbolic Tools

Theory, part 119

Practice, part 120

Key terms

Undetermined coefficients in antiderivatives

Differential binomials

Reduction strategies

Tables of integrals

Computer algebra

Non-elementary antiderivatives

Differentiation check

Check yourself

How this lesson connects

Builds on