1 Integration of trigonometric functions

Simple integrals involving trigonometric functions have already been dealt with in Section 13.1. See what you can remember:

Task!

Write down the following integrals:

  1. ∫ sin x d x ,
  2. ∫ cos x d x ,
  3. ∫ sin 2 x d x ,
  4. ∫ cos 2 x d x
  1. − cos x + c ,
  2. sin x + c ,
  3. − 1 2 cos 2 x + c ,
  4. 1 2 sin 2 x + c .

The basic rules from which these results can be derived are summarised here:

Key Point 8
∫ sin k x d x = − cos k x k + c ∫ cos k x d x = sin k x k + c

In engineering applications it is often necessary to integrate functions involving powers of the trigonometric functions such as

∫ sin 2 x d x  or  ∫ cos 2 ω t d t

Note that these integrals cannot be obtained directly from the formulas in Key Point 8 above. However, by making use of trigonometric identities, the integrands can be re-written in an alternative form. It is often not clear which identities are useful and each case needs to be considered individually. Experience and practice are essential. Work through the following Task.

Task!

Use the trigonometric identity   sin 2 θ ≡ 1 2 ( 1 − cos 2 θ )  to express the integral ∫ sin 2 x d x in an alternative form and hence evaluate it.

  1. First use the identity:

    The integral can be written ∫ 1 2 ( 1 − cos 2 x ) d x .

    Note that the trigonometric identity is used to convert a power of sin x into a function involving cos 2 x which can be integrated directly using Key Point 8.

  2. Now evaluate the integral:

    1 2 x − 1 2 sin 2 x + c = 1 2 x − 1 4 sin 2 x + K where K = c ∕ 2 .

Task!

Use the trigonometric identity sin 2 x ≡ 2 sin x cos x to find ∫ sin x cos x d x

  1. First use the identity:

    The integrand can be written as 1 2 sin 2 x

  2. Now evaluate the integral:

    ∫ 0 2 π sin x cos x d x = ∫ 0 2 π 1 2 sin 2 x d x = − 1 4 cos 2 x + c 0 2 π = − 1 4 cos 4 π + 1 4 cos 0 = − 1 4 + 1 4 = 0

    This result is one example of what are called orthogonality relations .