L7-L8: Trigonometric Integrals

Lecture notes covering techniques for integrating powers and products of trigonometric functions. The overarching goal is to rewrite the integrand into a form that can be integrated using standard rules, substitution, or integration by parts.

Rewriting Tools: Trigonometric Identities

Reciprocal & Quotient Identities

$\sin x = \frac{1}{\csc x}$, $\csc x = \frac{1}{\sin x}$
$\cos x = \frac{1}{\sec x}$, $\sec x = \frac{1}{\cos x}$
$\tan x = \frac{1}{\cot x} = \frac{\sin x}{\cos x}$
$\cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x}$

Pythagorean Identities

$\sin^2 x + \cos^2 x = 1$
$1 + \tan^2 x = \sec^2 x$
$1 + \cot^2 x = \csc^2 x$

Double-Angle Formulas

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

Power-Reducing (Half-Angle) Formulas

$\sin^2 u = \frac{1 - \cos(2u)}{2}$
$\cos^2 u = \frac{1 + \cos(2u)}{2}$
$\tan^2 u = \frac{1 - \cos(2u)}{1 + \cos(2u)}$

General Guidelines for $\int \sin^n x \cos^m x,dx$

Case	Strategy
$n$ odd	Strip one $\sin x$, convert remaining even sines to cosines using $\sin^2 x = 1 - \cos^2 x$, then substitute $u = \cos x$.
$m$ odd	Strip one $\cos x$, convert remaining even cosines to sines using $\cos^2 x = 1 - \sin^2 x$, then substitute $u = \sin x$.
Both odd	Use either of the above strategies.
Both even	Use power-reducing / double-angle formulas to reduce to integrable terms. Count zero as even.

flowchart TD
    A(["∫ sin^n x cos^m x dx"]) --> B{n odd?}
    B -->|Yes| C["Strip one sin x<br/>Convert rest to cos with<br/>sin²x = 1 - cos²x<br/>Substitute u = cos x"]
    B -->|No| D{m odd?}
    D -->|Yes| E["Strip one cos x<br/>Convert rest to sin with<br/>cos²x = 1 - sin²x<br/>Substitute u = sin x"]
    D -->|No| F["Both even<br/>Use power-reducing / double-angle formulas"]
    C --> G([Integrate])
    E --> G
    F --> G

    H(["∫ tan^m x sec^n x dx"]) --> I{n even?}
    I -->|Yes| J["Split sec²x<br/>Replace sec²x = 1 + tan²x<br/>Substitute u = tan x"]
    I -->|No| K{m odd?}
    K -->|Yes| L["Split sec x tan x<br/>Replace tan²x = sec²x - 1<br/>Substitute u = sec x"]
    K -->|No| M["m even, n odd<br/>Reduce to sec^n x alone<br/>Use reduction formula"]
    J --> N([Integrate])
    L --> N
    M --> N

Examples

Example 1 — Odd power of sine $$\int \sin^3 x,dx = \int (1 - \cos^2 x)\sin x,dx \xrightarrow{u=\cos x} -\cos x + \frac{1}{3}\cos^3 x + C$$

Example 2 — Odd power of cosine $$\int \sin^4 x \cos^5 x,dx = \int \sin^4 x (1 - \sin^2 x)^2 \cos x,dx \xrightarrow{u=\sin x} \frac{1}{5}\sin^5 x - \frac{2}{7}\sin^7 x + \frac{1}{9}\sin^9 x + C$$

Example 3 — Both even (repeated power-reducing) $$\int \sin^4 x,dx = \int\left(\frac{1 - \cos 2x}{2}\right)^2 dx = \frac{1}{4}\left[\frac{3}{2}x - \sin 2x + \frac{1}{8}\sin 4x\right] + C$$

Example 4 — Mixed even powers $$\int \sin^2 x \cos^4 x,dx = \frac{1}{8}\left[\frac{1}{2}x - \frac{1}{8}\sin 4x + \frac{1}{6}\sin^3 2x\right] + C$$

Integrating Powers of Tangent and Secant

Powers of Tangent

Basic result: $$\int \tan x,dx = \ln|\sec x| + C$$

For $\int \tan^m x,dx$ with $m > 1$:

Strip $\tan^2 x$: $\int \tan^{m-2} x ,\tan^2 x,dx$
Replace $\tan^2 x = \sec^2 x - 1$
Use substitution $u = \tan x$, $du = \sec^2 x,dx$ on the term containing $\sec^2 x$

Example 5 $$\int \tan^5 x,dx = \frac{\tan^4 x}{4} - \frac{\tan^2 x}{2} + \ln|\sec x| + C$$

Example 6 $$\int \tan^4 x,dx = \frac{\tan^3 x}{3} - \tan x + x + C$$

Powers of Secant

Basic result: $$\int \sec x,dx = \ln|\sec x + \tan x| + C, \qquad \int \sec^2 x,dx = \tan x + C$$

Even powers:

Factor out $\sec^2 x$: $\int \sec^m x,dx = \int \sec^2 x \cdot \sec^{m-2} x,dx$
Replace $\sec^2 x = 1 + \tan^2 x$
Substitute $u = \tan x$

Example 7 $$\int \sec^4 x,dx = \int (\tan^2 x + 1)\sec^2 x,dx = \frac{\tan^3 x}{3} + \tan x + C$$

Odd powers ($m \ge 3$): Use the reduction formula (or integration by parts): $$\int \sec^n x,dx = \frac{\sec^{n-2} x \tan x}{n-1} + \frac{n-2}{n-1}\int \sec^{n-2} x,dx$$

Example 8 — Integration by parts for $\int \sec^3 x,dx$ $$\int \sec^3 x,dx = \frac{1}{2}\sec x \tan x + \frac{1}{2}\ln|\sec x + \tan x| + C$$

Example 9 — Reduction formula for $\int \sec^5 x,dx$ $$\int \sec^5 x,dx = \frac{1}{4}\sec^3 x \tan x + \frac{3}{8}\sec x \tan x + \frac{3}{8}\ln|\sec x + \tan x| + C$$

Integrating Products $\int \tan^m x \sec^n x,dx$

Case	Strategy
$n$ even	Split off $\sec^2 x$, replace remaining $\sec^2 x$ with $1 + \tan^2 x$, substitute $u = \tan x$.
$m$ odd	Split off $\sec x \tan x$, replace $\tan^2 x$ with $\sec^2 x - 1$, substitute $u = \sec x$.
$m$ even, $n$ odd	Reduce to powers of $\sec x$ alone using $\tan^2 x = \sec^2 x - 1$, then use secant procedures (often integration by parts or reduction).

Example 10 — $n$ even $$\int \tan^2 x \sec^4 x,dx = \int (\tan^4 x + \tan^2 x)\sec^2 x,dx = \frac{\tan^5 x}{5} + \frac{\tan^3 x}{3} + C$$

Example 11 — $m$ odd $$\int \tan^3 x \sec^3 x,dx = \int (\sec^4 x - \sec^2 x)(\sec x \tan x),dx = \frac{\sec^5 x}{5} - \frac{\sec^3 x}{3} + C$$

Example 12 — $m$ even, $n$ odd $$\int \tan^2 x \sec x,dx = \int (\sec^3 x - \sec x),dx = \frac{1}{2}\sec x \tan x - \frac{1}{2}\ln|\sec x + \tan x| + C$$

Product-to-Sum Formulas

Used for integrals of the form $\int \sin mx \cos nx,dx$, $\int \sin mx \sin nx,dx$, $\int \cos mx \cos nx,dx$.

$\sin A \cos B = \frac{1}{2}\big[\sin(A-B) + \sin(A+B)\big]$
$\sin A \sin B = \frac{1}{2}\big[\cos(A-B) - \cos(A+B)\big]$
$\cos A \cos B = \frac{1}{2}\big[\cos(A-B) + \cos(A+B)\big]$

Example 13 $$\int \sin 2x \cos 3x,dx = \frac{1}{2}\int\big[-\sin x + \sin 5x\big],dx = \frac{1}{2}\cos x - \frac{1}{10}\cos 5x + C$$

Example 14 $$\int \sin 2x \sin 3x,dx = \frac{1}{2}\int\big[\cos x - \cos 5x\big],dx = \frac{1}{2}\sin x - \frac{1}{10}\sin 5x + C$$

Example 15 $$\int \cos 2x \cos 3x,dx = \frac{1}{2}\int\big[\cos x + \cos 5x\big],dx = \frac{1}{2}\sin x + \frac{1}{10}\sin 5x + C$$

Miscellaneous Algebraic Manipulations

Example 16 — Conjugate multiplication $$\int \frac{dx}{\cos x - 1} = \int \frac{\cos x + 1}{\cos^2 x - 1},dx = -\int\left(\frac{\cos x}{\sin^2 x} + \csc^2 x\right)dx = \csc x + \cot x + C$$

Example 17 — Simplifying before integrating $$\int \frac{1 - \tan^2 x}{\sec^2 x},dx = \int (\cos^2 x - \sin^2 x),dx = \int \cos 2x,dx = \frac{1}{2}\sin 2x + C$$

Example 18 — Substitution + power-reducing $$\int \cos^2(\sin x)\cos x,dx \xrightarrow{u=\sin x} \int \cos^2 u,du = \frac{1}{2}u + \frac{1}{4}\sin 2u + C = \frac{1}{2}\sin x + \frac{1}{4}\sin(2\sin x) + C$$

FAD1014 L7-L8 — Trigonometric Integrals