Trigonometric substitution and integration strategy

Lesson 1 introduced the four workhorse techniques of integration: substitution, integration by parts, trigonometric identities and partial fractions. Some common integrals still resist them, particularly those containing square roots of quadratic expressions such as \(\sqrt{a^2 - x^2}\) or \((x^2 + a^2)^{3/2}\). These appear whenever geometry involves circles, ellipses and distances: the area of a circular segment, the magnetic field of a straight wire, the electric field of a charged rod, the arc length of a parabola. This lesson adds trigonometric substitution, which turns such roots into trigonometric functions, along with completing the square, rationalizing substitutions and the integrals that lead to inverse hyperbolic functions. It ends with practical advice on integral tables, computer algebra systems and special functions, and a strategy for attacking any integral you meet.

The idea of trigonometric substitution

The Pythagorean identities \(1 - \sin^2\theta = \cos^2\theta\), \(1 + \tan^2\theta = \sec^2\theta\) and \(\sec^2\theta - 1 = \tan^2\theta\) each turn a difference or sum of squares into a single square. Substituting \(x\) as a trigonometric function of a new variable \(\theta\) can therefore remove a square root:

The restricted ranges of \(\theta\) make the substitutions one-to-one and ensure that the square roots are positive, so \(\sqrt{\cos^2\theta} = \cos\theta\) without an absolute value. After integrating in \(\theta\), convert back to \(x\) using a reference right triangle drawn from the substitution: for \(x = a\sin\theta\), the triangle has hypotenuse \(a\), opposite side \(x\), and adjacent side \(\sqrt{a^2 - x^2}\).

Roots of the form √(a² − x²)

Roots of the form √(a² + x²)

Integrals with \(a^2 + x^2\) arise whenever the distance from a point to a line is involved, since the distance from a point at height \(d\) above a line to a point \(x\) along it is \(\sqrt{x^2 + d^2}\). The tangent substitution handles them.

Integrals of \(1/\sqrt{x^2 + a^2}\) lead to logarithms. With \(x = a\tan\theta\), \(\int \frac{dx}{\sqrt{x^2 + a^2}} = \int \sec\theta\,d\theta = \ln|\sec\theta + \tan\theta| + C\), which converts back to:

The inverse hyperbolic sine, \(\operatorname{arsinh}\), is the inverse of the hyperbolic sine met in Calculus I (MATH-100), and the logarithmic form shows how hyperbolic functions are built from exponentials. Similarly, \(\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln\left|x + \sqrt{x^2 - a^2}\right| + C\).

Roots of the form √(x² − a²)

Completing the square

A quadratic \(ax^2 + bx + c\) that is not a pure sum or difference of squares can be brought into one of the standard forms by completing the square. Then a simple shift \(u = x + b/(2a)\) prepares it for a table entry or a trigonometric substitution.

When the numerator contains \(x\), split the integral into a part that is a multiple of the derivative of the quadratic (handled by substitution) and a constant part (handled as above). For example, \(\dfrac{2x + 7}{x^2 + 4x + 13} = \dfrac{2x + 4}{x^2 + 4x + 13} + \dfrac{3}{x^2 + 4x + 13}\), whose integrals are \(\ln(x^2 + 4x + 13)\) and \(\arctan\frac{x + 2}{3}\). This is also how the irreducible quadratic factors of the partial fraction method in Lesson 1 are integrated.

Rationalizing substitutions

Integrands containing a root of a linear expression, such as \(\sqrt{x}\) or \(\sqrt[3]{x + 1}\), can often be turned into rational functions by substituting \(u\) for the root.

A remarkable substitution, \(t = \tan(\theta/2)\), turns any rational function of \(\sin\theta\) and \(\cos\theta\) into a rational function of \(t\), since \(\sin\theta = \frac{2t}{1 + t^2}\), \(\cos\theta = \frac{1 - t^2}{1 + t^2}\) and \(d\theta = \frac{2\,dt}{1 + t^2}\). It is rarely the quickest method by hand, but it guarantees that such integrals can always be done, and computer algebra systems use it.

Tables, computer algebra and special functions

Engineers do not integrate everything by hand. Extensive tables of integrals have been published for over a century, and computer algebra systems, built into mathematical software and available online, can find antiderivatives of a vast range of functions. Their results can, however, look very different from a hand calculation (the same antiderivative may be written with logarithms, inverse hyperbolic functions or a different constant), may assume conditions such as positive parameters without saying so, and occasionally contain errors at points where expressions are undefined. A sound habit is to verify any antiderivative, from any source, by differentiating it.

Many integrals that matter in engineering have no elementary antiderivative at all, and are defined as special functions, tabulated and built into software like the familiar trigonometric functions:

• The error function, \(\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}}\int_0^x e^{-t^2}\,dt\), gives probabilities for the normal distribution (Lesson 6) and solutions of the heat equation, such as how far a temperature change penetrates into soil or concrete.
• Elliptic integrals give the exact period of a pendulum at large amplitude and the perimeter of an ellipse.
• The Fresnel integrals, \(\int \sin(t^2)\,dt\) and \(\int \cos(t^2)\,dt\), describe diffraction of light and also the transition curves (clothoids) used in road and railway design to change curvature smoothly between straight and curved sections.

A strategy for integration

Combining Lesson 1 with this lesson, a practical order of attack for an unfamiliar integral is:

• Simplify: expand, cancel, split fractions, apply identities.
• Look for a standard form, perhaps after a constant adjustment or completing the square.
• Look for a function and its derivative: substitution.
• Products of different kinds of function: integration by parts.
• Powers and products of trigonometric functions: identities.
• Rational functions: polynomial division, then partial fractions.
• Square roots of quadratics: complete the square, then trigonometric substitution.
• Roots of linear expressions: rationalizing substitution.
• If all else fails: tables or computer algebra (and verify), special functions, or numerical integration.

The skill lies less in memorizing every case than in recognizing which form an integral can be turned into. That recognition comes from practice, which is why the practice problems at the end of each lesson matter.