Integration Using Trigonometric Identities 34

1. Core Concept Overview & Intuition

Integration is defined as the reverse or converse process of differentiation. In this unit the integrand is a product, power, or rational expression built from trigonometric functions, and the strategy is to rewrite it, using trigonometric identities, into a form that matches one of the basic formulae already established. The basic trigonometric formulae available are:

$$
\int \cos x\,dx = \sin x + c
$$

$$
\int \sin x\,dx = -\cos x + c
$$

$$
\int \sec^{2} x\,dx = \tan x + c
$$

$$
\int \sec x \tan x\,dx = \sec x + c
$$

$$
\int \csc x \cot x\,dx = -\csc x + c
$$

The intuition is uniform across the whole chapter: an integrand such as $\sin^{2} x$ or $\sin 3x \cos 2x$ is not immediately integrable by a basic formula, but an identity converts it into a sum of terms each of which is a constant multiple of $\sin(kx)$, $\cos(kx)$, or $\sec^{2}(kx)$. Once in that form, the extension formulae and the basic formulae finish the job. The constant of integration $c$ is always appended, because any constant added to an antiderivative disappears under differentiation.

2. Mathematical Formalization & Core Definitions

> Definition (Integrand). If $f(x)$ is a function of $x$, then $\int f(x)\,dx$ denotes the integral of $f(x)$ with respect to $x$. The function $f(x)$ is called the integrand. The integral sign cannot be divorced from $dx$ when the integration is with respect to $x$.

> Definition (Constant of Integration). Since $\frac{d}{dx}(\sin x) = \frac{d}{dx}(\sin x + c) = \cos x$ for any constant $c$, every indefinite integral is written with an appended constant $c$, called the constant of integration.

The extension formulae (from the earlier study of extensions of the basic formulae) permit integration of a trigonometric function of a linear argument $u = ax + b$:

$$
\int \cos(ax+b)\,dx = \frac{1}{a}\sin(ax+b) + c
$$

$$
\int \sin(ax+b)\,dx = -\frac{1}{a}\cos(ax+b) + c
$$

$$
\int \sec^{2}(ax+b)\,dx = \frac{1}{a}\tan(ax+b) + c
$$

Review of Important Trigonometric Identities

The identities used throughout this unit are grouped as follows.

The three fundamental identities.

$$
\sin^{2} x + \cos^{2} x = 1
$$

$$
1 + \tan^{2} x = \sec^{2} x
$$

$$
1 + \cot^{2} x = \csc^{2} x
$$

The formulae for $\cos 2x$.

$$
\cos 2x = \cos^{2} x - \sin^{2} x
$$

$$
\cos 2x = 2\cos^{2} x - 1
$$

$$
\cos 2x = 1 - 2\sin^{2} x
$$

Rearranged, the last two give the power-reducing forms:

$$
\cos^{2} x = \frac{1 + \cos 2x}{2}
$$

$$
\sin^{2} x = \frac{1 - \cos 2x}{2}
$$

The product formulae.

$$
2\sin A \cos B = \sin(A+B) + \sin(A-B)
$$

$$
2\cos A \sin B = \sin(A+B) - \sin(A-B)
$$

$$
2\cos A \cos B = \cos(A+B) + \cos(A-B)
$$

$$
2\sin A \sin B = \cos(A-B) - \cos(A+B)
$$

The formulae for $\sin 3x$ and $\cos 3x$.

$$
\sin 3x = 3\sin x - 4\sin^{3} x
$$

$$
\cos 3x = 4\cos^{3} x - 3\cos x
$$

Rearranged, these give the power-reducing forms for cubes:

$$
\sin^{3} x = \frac{3\sin x - \sin 3x}{4}
$$

$$
\cos^{3} x = \frac{3\cos x + \cos 3x}{4}
$$