Trigonometric Integrals

From the basic integrals of $\int \sin x dx=-\cos x+C$ and $\int\cos x dx=\sin x+C$, we can compute the indefinite integrals of many trigonometric functions, using a combination of trigonometric identities and integration techniques (for now, substitution and parts). This lecture will not introduce any fundamentally new techniques, but rather will help tie closer together our knowledge bases in trigonometry and calculus.

Trigonometric Identities

Its useful to begin by reviewing some trigonometric identities that will aid us in converting trigonometric expressions from one form to another. The most fundamental identity is the pythagorean theorem:

$$\sin^2\theta+\cos^2\theta=1$$

For us, this will represent a means of converting one trigonometric function into another, replacing say $\sin^2 x$ with $1-\cos^2 x$ when needed. Dividing this identity through by $\cos^2$ gives a relationship between $\sec$ and $\tan$ $$\sec^2\theta = \tan^2\theta+1$$ And dividing instead by $\sin^2$ gives an analogous relationship between $\csc$ and $\cot$. Other identities that prove useful are the double angle identities, which tell us how to express the basic trigonometric functions of $2\theta$ in terms of trigonometric functions of $\theta$:

$$\sin(2\theta)=2\sin\theta\cos\theta$$ $$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

It’s occasionally useful also to recall the half angle identities (which can be derived from all thats above), allowing us to write a trigonometric function of $\theta$ in terms of $2\theta$ (this may also be written as $\theta/2$ in terms of $\theta$…hence the name)

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

Trigonometric Derivatives

Its probably also helpful to review some of the differential calculus of trigonometric functions. In one sense, this is all unnecessary since once we know $(\sin)^\prime=\cos$ and $(\cos)^\prime=-\sin$, everything else is easily derivable using the basic techniques of differentiation (the chain rule, product rule and quotient rule). However, its sometimes useful to have a couple others on easy recall:

$$\frac{d}{d\theta}\tan\theta = \sec^2\theta$$ $$\frac{d}{d\theta}\cot\theta = -\csc^2\theta$$ $$\frac{d}{d\theta}\sec\theta = \sec\theta\tan\theta$$ $$\frac{d}{d\theta}\csc\theta = -\csc\theta\cot\theta$$

Integration Examples

Here are a few special cases of trigonometric functions whose antiderivatives can be quickly found using the above facts: The main idea of all of these is as follows: we want to do a $u$-sub to remove the trigonometry. To do so, we need to choose a $u$ and find $du$ in the integral. So, we should choose a differential trig identity to relate $du$ to the trig functions that appear, and then use standard trig identites to convert the rest of the integral into $u$.

Products of Sine and Cosine

$$\int\cos^3\theta d\theta$$ $$\int\cos^2(t)\sin^3(t)dt$$ $$\int(1+\cos(t))\sin^5(t)dt$$

In general, when an integral contains a product of copies of $\sin$ and $\cos$, try to use $\sin^2+\cos^2=1$ to convert all but one of the terms to $\cos$, leaving a single $\sin$ (or vice versa), and then perform a u-substitution. What happens when you can’t separate out a single $\sin$ or $\cos$?

$$\int\sin^2(x)dx$$

A little more involved, involving using the half angle identities twice: $$\int\sin^4(x)dx$$

Powers of Secant and Tangent

One way to deal with secant, tangent and other trigonometric functions is to try to reduce them to their definitions in terms of sine and cosine before proceeding. For example,

$$\int\tan(x)dx$$

But additionally, the relationships between $\tan$ and $\sec$ through differentiation allow us also to integrate many different combinations of these basic trigonometric functions:

$$\int \tan^6 x\sec^4 xdx$$ Here, separate out one factor of $\sec^2$ (the derivative of tangent), and convert the rest of the secants into tangents using $\sec^2=\tan^2+1$ before performing a u-substitution.

$$\int tan^5\theta\sec^7\theta d\theta$$ Here, we can’t separate out a $\sec^2$ without leaving an odd number of secants in the integral (which don’t convert nicely to tangents!) Instead, separate out a $\sec x\tan x$ (the derivative of $\sec x$) and convert everything else into secants (again using $\sec^2=\tan^2+1$).

The Integral of Secant

The antiderivative of the secant function is something we can calculate using some very clever manipulations and a u-substituion (see Section 7.2 of the textbook), but we will hold off on deriving this in class until we have a more streamlined method available (partial fractions).

For now, if you need it; here’s the antiderivative $$\int\sec(x)dx=\ln|\sec x+\tan x|+C$$ You should verify this is correct by taking the derivative of both sides!