Trigonometric Integrals

Trigonometric identities:

sin²x + cos²x = 1

sin²x =   ( 1 – cos 2x)

cos² x = ( 1 + cos 2x)

sec²x = 1+ tan²x

Definition:

A function F(x) is called an anti derivative or integral of a function f(x) on an interval I if

F’(x) = f(x) for every value of x in I.

The symbol ‘ ∫ ’ is the sign of integration.

The function f(x) is called Integrand.

The variable x in dx is called variable of integration or integrator.

The process of finding the integral is called integration.

Trigonometric integration:

Integrals of products of Sines and Cosines:

Example:

Put u=sinx, du=cosx dx

Integrals of Secants and Tangents:

More generally an integral of the form

∫ tan^mx Secⁿx dx ,can be computed in the following way:

• if m is odd, use u=secx, du= secx tanx dx
• if n is even, use u= tan x, du=  sec²xdx
• if m is odd and n is even it doesn’t matter which method we use, so use first one(1).

To evaluate THE integrals (a) ∫sin mxcos nx dx, (b) ∫sin mxsin nx dx , (c) ∫cos mxcos nx dx use the corresponding identity:

corresponding identity:

• sinA cosB = ½ [sin (A-B) + sin (A+B)]
• sin A sin B = ½ [cos ( A- B) – cos ( A + B )]

Trigonometric substitutions:

The following substitutions are useful in integrals containing the following expressions:

Example: