Many trig expressions are not standard integrals and must be rewritten with an identity first. Use the double angle formulae for squares and products: sin²x = (1 − cos 2x)/2, cos²x = (1 + cos 2x)/2 and sin x cos x = ½ sin 2x. Use the Pythagorean identities for tan²x = sec²x − 1 and cot²x = cosec²x − 1. After rewriting, each term is a standard integral.
Identities for integrating trig
- sin²x = (1 − cos 2x)/2
- cos²x = (1 + cos 2x)/2
- sin x cos x = ½ sin 2x
- tan²x = sec²x − 1
- cot²x = cosec²x − 1
∫ cos kx dx = (1/k)sin kx + c and ∫ sin kx dx = −(1/k)cos kx + c.
Integrating sin²x and cos²x
You cannot integrate sin²x or cos²x directly; there is no standard result for a squared trig function. Replace each one using a rearranged double angle formula: sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2. Every term is then a constant or a cosine of a multiple angle.
When you integrate cos 2x, remember to divide by 2. For a square of a multiple angle, such as sin²2x, the identity gives cos 4x, so you divide by 4.
Products of sin x and cos x
A product of sine and cosine of the same angle collapses with the double angle formula sin 2x = 2 sin x cos x, so sin x cos x = ½ sin 2x. Spotting that pattern turns a product into a single sine term that integrates easily.
Integrating tan²x and cot²x
These have no standard integral either. Rearrange the Pythagorean identities so that sec² or cosec² appears: tan²x = sec²x − 1 and cot²x = cosec²x − 1. Then ∫ sec²x dx = tan x and ∫ cosec²x dx = −cot x. The −1 integrates to −x; don't drop it.
Combining identities
Harder integrands need more than one step. Expand a bracket, spot a difference such as cos²x − sin²x = cos 2x, or use a double angle formula twice. The aim is always a sum of terms of the form a constant, cos kx or sin kx, each of which can be integrated directly.
Exact areas with trig
After rewriting and integrating, evaluate between the limits. At x = 0, π and 2π the sine terms are zero, so areas involving squared trig functions often come out as exact multiples of π. Give answers exactly, such as π/8 + 1/4, rather than as decimals.
Worked examples
Example 1. Find ∫ sin²2x dx.
- With 2x as the angle: sin²2x = (1 − cos 4x)/2.
- ½ ∫ (1 − cos 4x) dx = ½(x − ¼ sin 4x).
Answer: ½x − ⅛ sin 4x + c.
Marks: M1 for the double angle identity, M1 for integrating cos 4x, A1 for the answer.
Example 2. Find ∫ 3 sin x cos x dx.
- Rewrite: 3 sin x cos x = (3/2) sin 2x.
- Integrate: (3/2) × (−½ cos 2x).
Answer: −¾ cos 2x + c.
Marks: M1 for (3/2) sin 2x, A1 for −¾ cos 2x + c.
Example 3. Find ∫ 2 tan²x dx.
- Rewrite: 2 tan²x = 2(sec²x − 1).
- Integrate: 2 tan x − 2x.
Answer: 2 tan x − 2x + c.
Marks: M1 for tan²x = sec²x − 1, A1 for the answer.
Example 4. Find ∫ (1 − cos x)² dx.
- Expand: 1 − 2 cos x + cos²x.
- Rewrite cos²x = (1 + cos 2x)/2, giving 3/2 − 2 cos x + ½ cos 2x.
- Integrate term by term.
Answer: (3/2)x − 2 sin x + ¼ sin 2x + c.
Marks: M1 for expanding, M1 for the cos²x identity, A1 for the answer.
Example 5. Find the exact value of ∫ cos²x dx from 0 to π/4.
- cos²x = (1 + cos 2x)/2, which integrates to ½x + ¼ sin 2x.
- Evaluate: (π/8 + ¼ sin(π/2)) − 0.
Answer: π/8 + 1/4.
Marks: M1 for the identity and integration, A1 for π/8 + 1/4.
Common mistakes
- Integrating sin²x as a power. sin²x is not a power of x; rewrite it with a double angle identity first.
- Mixing up the signs in the identities. sin²x = (1 − cos 2x)/2 has the minus; cos²x = (1 + cos 2x)/2 has the plus.
- Not dividing by 2 when integrating cos 2x. ∫ cos 2x dx = ½ sin 2x + c.
- Dropping the −1 from tan²x = sec²x − 1. It integrates to −x, which must appear in the answer.
- Rounding exact answers. Leave answers such as π/8 + 1/4 in exact form.
How trig integration is examined
Integrating with trig identities is examined on the Pure 3 paper (WMA13) for Edexcel International A-Level, and in Year 2 Pure of Edexcel A-Level (9MA0). A question usually:
- asks you to write sin²x, cos²x or tan²x in a form you can integrate
- then asks for the integral, using “hence”
- finds an exact area, often in terms of π
- combines an expansion with more than one identity
Multi-part questions of this kind, with mark schemes, are among the Pure 3 integration exam questions. The Pure 3 integration worksheet and the integration revision engine give extra practice.
The identities used here come from the double angle formulae and sec, cosec and cot. The same technique appears in volumes of revolution.
Frequently asked questions
How do you integrate sin²x?
Use sin²x = (1 − cos 2x)/2. Then ∫ sin²x dx = ½x − ¼ sin 2x + c.
How do you integrate tan²x?
Write tan²x = sec²x − 1. Then ∫ tan²x dx = tan x − x + c.
How do you integrate sin x cos x?
Write it as ½ sin 2x, which integrates to −¼ cos 2x + c.