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Integrating Trigonometric Functions Using Identities

This lesson shows how to integrate trigonometric functions that basic rules cannot handle directly by first rewriting the integrand with Pythagorean, double-angle, or power-reducing identities, then applying standard integration rules or a simple substitution.

Why some trig integrals need an identity first

The basic integration rules only cover a handful of trig antiderivatives directly, such as \(\int \sin x\,dx = -\cos x + C\) or \(\int \sec^2 x\,dx = \tan x + C\). As soon as the integrand involves a power of sine or cosine, a product of two different trig functions, or something like \(\tan^2 x\), none of the basic rules apply as-is. The fix is almost always the same: use a trigonometric identity to rewrite the integrand into a form that a basic rule (or a simple \(u\)-substitution) can handle.

This is a different skill from trig substitution integrals, where you introduce a trig function to simplify an algebraic expression like \(\sqrt{a^2 - x^2}\). Here, the integrand already is trigonometric, and the identity is used to simplify it, not to substitute a new variable.

Identities you will use most often

Identity family	Formula	Typical use
Pythagorean	\(\sin^2 x + \cos^2 x = 1\)	Odd powers of sine or cosine
Reciprocal / quotient	\(\tan^2 x = \sec^2 x - 1\), \(\cot^2 x = \csc^2 x - 1\)	Powers of tangent or cotangent
Power-reducing	\(\sin^2 x = \frac{1-\cos 2x}{2}\), \(\cos^2 x = \frac{1+\cos 2x}{2}\)	Even powers of sine or cosine
Double-angle	\(\sin 2x = 2\sin x \cos x\)	Products of sine and cosine

The power-reducing formulas come directly from the double angle identities, and the tangent/cotangent rewrites come from the quotient identities and reciprocal identities. Knowing those two families well makes almost every integral in this lesson mechanical.

Odd powers of sine or cosine: peel off one factor

When the power on sine or cosine is odd, split off a single factor and convert the rest with \(\sin^2 x = 1 - \cos^2 x\) (or the cosine version), then substitute.

Example 1: Evaluate \(\int \sin^3 x\,dx\).

Write \(\sin^3 x = \sin^2 x \cdot \sin x = (1-\cos^2 x)\sin x\). Let \(u = \cos x\), so \(du = -\sin x\,dx\):

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

Even powers of sine or cosine: reduce the power first

When the power is even, an odd-power split does not help because there is no leftover single factor to substitute for. Instead, use the power-reducing formulas to trade the square for a double-angle expression.

Example 2: Evaluate \(\int \sin^2 x\,dx\).

\(\int \sin^2 x\,dx = \int \frac{1-\cos 2x}{2}\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C\)

The graph below shows \(y = \sin^2 x\); notice it oscillates between 0 and 1 with period \(\pi\), which is exactly what the \(\cos 2x\) term in the antiderivative reflects.

Graph of y = sine squared x over one full period

Products of sine and cosine with matching or related angles

Example 3: Evaluate \(\int \sin^2 x \cos^2 x\,dx\).

Rather than reducing each factor separately, notice that \(\sin x \cos x = \frac{1}{2}\sin 2x\), so \(\sin^2 x \cos^2 x = \frac{1}{4}\sin^2 2x\). Apply the power-reducing formula again, this time to \(\sin^2 2x\):

\(\frac{1}{4}\sin^2 2x = \frac{1}{4}\cdot\frac{1-\cos 4x}{2} = \frac{1}{8} - \frac{\cos 4x}{8}\)

So \(\int \sin^2 x \cos^2 x\,dx = \frac{x}{8} - \frac{\sin 4x}{32} + C\).

Powers of tangent and secant

Example 4: Evaluate \(\int \tan^2 x\,dx\).

Since \(\tan^2 x = \sec^2 x - 1\), the integral becomes \(\int (\sec^2 x - 1)\,dx = \tan x - x + C\). Higher powers of tangent combined with secant usually split off a \(\sec^2 x\) factor for a \(u = \tan x\) substitution, using the same reciprocal identity to convert the remaining tangent factors.

Choosing the right strategy

A quick checklist helps you decide which identity to reach for:

Odd power of sine or cosine: peel off one factor, convert the rest with the Pythagorean identity, then substitute.
Even power of sine or cosine only: apply the power-reducing (half-angle) formulas.
Powers of tangent or cotangent: rewrite using \(\tan^2 x = \sec^2 x - 1\) or \(\cot^2 x = \csc^2 x - 1\).
Product of sine and cosine with different angle multiples: use product-to-sum identities to split the product into a sum of single sine or cosine terms.

If the integrand does not simplify with an identity and instead contains an algebraic expression like \(\sqrt{a^2-x^2}\) or \(a^2+x^2\), that is a signal to use trig substitution integrals instead. And if no elementary antiderivative exists at all, numerical integration methods give an approximate value.

Practice tip

Before integrating anything trigonometric, ask "can I rewrite this with an identity so a basic rule applies?" first. Getting comfortable recognizing odd versus even powers, and remembering the power-reducing and reciprocal identities, turns a page of intimidating integrals into a short, repeatable checklist.

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