Trigonometric Integrals

What You’ll Learn

In this lesson you’ll learn strategies for integrating products and powers of sine, cosine, tangent, and secant. These come up constantly in physics, engineering, and signal processing.

The Concept

Trigonometric integrals follow a few predictable patterns. Once you recognize which pattern you’re looking at, the strategy is straightforward.

Powers of sine and cosine

The key identity is the Pythagorean identity:

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

If at least one power is odd, peel off one factor for du and convert the rest using the identity above.

If both powers are even, use the power-reducing (half-angle) identities:

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

These convert even powers into expressions with double angles that you can integrate directly.

Powers of tangent and secant

The key identity here is:

$$\tan^2 x = \sec^2 x - 1$$

Combined with the fact that the derivative of tan(x) is sec²(x) and the derivative of sec(x) is sec(x)tan(x), you get two main strategies:

• If the power of secant is even: save sec²(x) for du, convert the rest to tangent
• If the power of tangent is odd: save sec(x)tan(x) for du, convert the rest to secant

Quick reference

Here’s the decision process:

1. Look at the powers of sine/cosine (or tangent/secant)
2. Is one power odd? Peel off one factor, convert the rest with the Pythagorean identity, substitute
3. Are both powers even? Use half-angle identities to reduce
4. Does nothing fit? Try rewriting everything in terms of sine and cosine first

Worked Example

Example 1: Odd power of sine

Evaluate:

$$\int \sin^3 x \, dx$$

Since the power of sine is odd (3), peel off one sin(x) for du and convert the rest:

$$\int \sin^2 x \cdot \sin x \, dx = \int (1 - \cos^2 x) \sin x \, dx$$

Let u = cos(x), du = -sin(x) dx:

$$-\int (1 - u^2) \, du = -(u - \tfrac{u^3}{3}) + C$$ $$= -\cos x + \frac{\cos^3 x}{3} + C$$

Example 2: Both powers even

Evaluate:

$$\int \sin^2 x \cos^2 x \, dx$$

Both powers are even, so apply the half-angle identities:

$$\sin^2 x \cos^2 x = \frac{1 - \cos 2x}{2} \cdot \frac{1 + \cos 2x}{2} = \frac{1 - \cos^2 2x}{4}$$

Apply the half-angle identity again to cos²(2x):

$$= \frac{1}{4} - \frac{\cos^2 2x}{4} = \frac{1}{4} - \frac{1}{4} \cdot \frac{1 + \cos 4x}{2} = \frac{1}{4} - \frac{1}{8} - \frac{\cos 4x}{8}$$ $$= \frac{1}{8} - \frac{\cos 4x}{8} = \frac{1 - \cos 4x}{8}$$

Now integrate:

$$\int \frac{1 - \cos 4x}{8} \, dx = \frac{1}{8}\left(x - \frac{\sin 4x}{4}\right) + C = \frac{x}{8} - \frac{\sin 4x}{32} + C$$

Example 3: Tangent and secant

Evaluate:

$$\int \tan^3 x \sec^2 x \, dx$$

The derivative of tan(x) is sec²(x), so this is a direct substitution.

Let u = tan(x), du = sec²(x) dx:

$$\int u^3 \, du = \frac{u^4}{4} + C = \frac{\tan^4 x}{4} + C$$

Real-World Application

Trigonometric integrals show up anywhere periodic functions are involved:

• In electrical engineering, computing power in AC circuits requires integrating products of sine and cosine
• In physics, energy calculations for oscillating systems (springs, pendulums) involve these integrals
• In audio and signal processing, Fourier analysis decomposes signals into sine and cosine components, and computing coefficients requires exactly these types of integrals
• In game development, smooth periodic animations and wave effects use trig functions, and integrating them gives accumulated displacement or energy

You’ve Got This

The patterns here are very consistent. Odd power? Peel one off, convert the rest, substitute. Both even? Half-angle identities. Tangent and secant? Check if sec²(x) or sec(x)tan(x) is sitting there ready to be du. You don’t need to memorize every case. Just ask: “Is there an odd power I can exploit?” If yes, you know what to do. If no, reach for the half-angle formulas. That’s really it.

Quiz

When integrating sin³(x), the best first step is to:

• A.Use the half-angle identity
• B.Apply integration by parts
• C.Save one sin(x) for du and write sin²(x) as 1 - cos²(x)
• D.Substitute u = sin(x) directly

For integrals with even powers of both sine and cosine, you should use:

• A.U-substitution with u = sin(x)
• B.Power-reducing (half-angle) identities
• C.Integration by parts
• D.The Pythagorean identity directly

The integral of sin²(x) cos²(x) dx simplifies to:

• A.(1/4)sin²(2x) + C
• B.x/8 + sin(4x)/32 + C
• C.x/8 - sin(4x)/32 + C
• D.(1/2)x - (1/4)sin(2x) + C

For the integral of tan³(x) sec²(x) dx, the best substitution is:

• A.u = sec(x)
• B.u = sin(x)
• C.u = cos(x)
• D.u = tan(x)

Which identity is most useful for converting between tangent and secant?

• A.sin²(x) + cos²(x) = 1
• B.sin(2x) = 2 sin(x) cos(x)
• C.tan²(x) = sec²(x) - 1
• D.cos(2x) = 1 - 2sin²(x)