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11.3PureStretch

Using Trigonometric Identities

Many trig expressions cannot be integrated directly. The trick is to rewrite them with an identity — especially the double-angle forms — into something with a standard integral.

24 min Integration
Edexcel A level Maths: 11.3 Using Trigonometric Identities To IntegrateCurated by Kepler, not produced by us. Watch it before the notes below.
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What you'll be able to do

  • Integrate sin²x and cos²x
  • Use double-angle identities
  • Rewrite products into integrable forms
  • Apply to definite integrals
1

Power-reduction

You cannot integrate cos⁡2x\cos^2 x directly, but cos⁡2x=12(1+cos⁡2x)\cos^2 x = \tfrac12(1 + \cos 2x) integrates easily. Similarly sin⁡2x=12(1−cos⁡2x)\sin^2 x = \tfrac12(1 - \cos 2x).

cos⁡2x=12(1+cos⁡2x),sin⁡2x=12(1−cos⁡2x)\cos^2 x = \tfrac12(1 + \cos 2x), \quad \sin^2 x = \tfrac12(1 - \cos 2x)
Rewrite before integrating.
Example 1Find ∫cos⁡2x dx\int \cos^2 x\,dx.
1=∫12(1+cos⁡2x) dx= \int \tfrac12(1 + \cos 2x)\,dx.
2=12x+14sin⁡2x+c= \tfrac12 x + \tfrac14\sin 2x + c.
Answer
12x+14sin⁡2x+c\tfrac12 x + \tfrac14\sin 2x + c

Tip — See a squared sin or cos? Reach for a double-angle identity first.

Formula recap

∫cos⁡2x dx=12x+14sin⁡2x+c\int \cos^2 x\,dx = \tfrac12 x + \tfrac14\sin 2x + c
Via cos² identity.
∫sin⁡2x dx=12x−14sin⁡2x+c\int \sin^2 x\,dx = \tfrac12 x - \tfrac14\sin 2x + c
Via sin² identity.

Common mistakes to avoid

Integrating cos²x to (cos³x)/3 type answers.
Use the identity first; there is no simple power rule for cos²x.
Forgetting the 1/a factor when integrating cos 2x.
∫cos 2x dx = ½ sin 2x.

Key takeaways

  • cos²x = ½(1+cos2x); sin²x = ½(1−cos2x).
  • Rewrite with an identity, then integrate term by term.
  • Mind the 1/a factor for cos(2x).

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. To integrate cos⁡2x\cos^2 x you first write it as:

    • 12(1+cos⁡2x)\tfrac12(1+\cos 2x)
    • 12(1−cos⁡2x)\tfrac12(1-\cos 2x)
    • 1−sin⁡2x1-\sin^2 x
    • 2cos⁡x2\cos x
    Show answer and explanation

    Answer: 12(1+cos⁡2x)\tfrac12(1+\cos 2x)

    Power reduction.

  2. ∫cos⁡2x dx\int \cos^2 x\,dx equals:

    • 12x+14sin⁡2x+c\tfrac12 x + \tfrac14\sin 2x + c
    • 13cos⁡3x+c\tfrac13\cos^3 x + c
    • 12x−14sin⁡2x+c\tfrac12 x - \tfrac14\sin 2x + c
    • sin⁡2x+c\sin 2x + c
    Show answer and explanation

    Answer: 12x+14sin⁡2x+c\tfrac12 x + \tfrac14\sin 2x + c

    Integrate the identity.

  3. sin⁡2x\sin^2 x rewrites as:

    • 12(1−cos⁡2x)\tfrac12(1-\cos 2x)
    • 12(1+cos⁡2x)\tfrac12(1+\cos 2x)
    • 1+cos⁡2x1+\cos^2 x
    • 2sin⁡x2\sin x
    Show answer and explanation

    Answer: 12(1−cos⁡2x)\tfrac12(1-\cos 2x)

    From 1−2sin².

  4. ∫cos⁡2x dx\int \cos 2x\,dx equals:

    • 12sin⁡2x+c\tfrac12\sin 2x + c
    • 2sin⁡2x+c2\sin 2x + c
    • sin⁡2x+c\sin 2x + c
    • −12sin⁡2x+c-\tfrac12\sin 2x + c
    Show answer and explanation

    Answer: 12sin⁡2x+c\tfrac12\sin 2x + c

    Divide by 2.

  5. Seeing sin²x or cos²x, you should:

    • use a double-angle identity
    • use the power rule
    • differentiate
    • substitute u = x²
    Show answer and explanation

    Answer: use a double-angle identity

    Then it integrates.

Next step

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