Integration of Trigonometric Functions

Equations_and_relations.jpgThe integration of trigonometric functions can be done by inspection.

The table below shows the formulae for the differentiation and integration of the basic trig. functions given to you for the Bursary examination.

y = f(x)


f '(x) or y ' or Y12_Integration_of_Trigonometric_Functions_01.gif
sin x
cos x
cos x
- sin x
tan x
sec 2x
sec x
sec x tan x
cosec x
- cosec x cot x
cot x
-cosec 2x

This means that if f(x) = sin x then f '(x) = cos x and therefore cos x = sin x.

Integrating trigonometric functions by inspection

This is in effect a trial and error method, similar to that used with exponential functions, but becomes quite easy with practice.

To find  sin 5x dx the answer could be guessed at cos 5x. However on differentiating cos 5x using the Chain 
Rule we get -5sin 5x which is -5 times too big so to compensate, a factor of -1 / 5 is required.

 sin 5x dx = Y12_Integration_of_Trigonometric_Functions_03.gif

Further example

 3sin 4x dx = Y12_Integration_of_Trigonometric_Functions_04.gif

Integrating trigonometric functions by substitution

Some people prefer a longer but more methodical method and the following substitution can be used.

Find  sin 5x dx

Let y =  sin 5x dx

and let u = 5x so y = sin u dx

"You may prefer the trial and error method!"Y12_Integration_of_Trigonometric_Functions_06.gif



Integration of Trigonometric Products

Sometimes trigonometric identities are useful in integration, such as the product formulae, covered in the Trigonometry section in Topic 43, Angle Formulae.

These formulae are:

2sin A cos B = sin(A + B) + sin(A − B)

2cos A sin B = sin(A + B) − sin(A − B)
2cos A cos B = cos(A + B) + cos(A − B)
2sin A sin B = cos(A − B) − cos(A + B)


Find  sin 3x cos 2x dx