This section will examine integration of trigonometric functions using basic trigonometric identities. Also, three types of trigonometric substitutions are introduced to help integrate complex trigonometric functions.

To start, the basic sine and cosine functions need to be integrated. In a previous section, the derivatives of trigonometric functions were introduced. For example,

      d(sinx)/dx = cosx

Integrating both side of the above equation gives,

The above equation means the integral of cosine is sine. In the similar way, the integral of sine is negative cosine.

Similarly, other basic trigonometric functions can also be integrated to give:

The above basic trigonometric identities are used to integrate certain combinations of trigonometric functions. The first type of combinations of trigonometric functions is powers of sine and cosine.

Form 1:

The following general strategy is used in evaluating integrals of the form , where are integers.

• If n is odd (n = 2k + 1, where k is integer), make the substitution cos²x = 1 - sin²x.
  
  
  
  The movie on the left gives an example of integrating sin²xcos³x.

• If m is odd (n = 2k + 1, where k is integer), make the substitution sin²x = 1 - cos²x.
  
  
  The diagram on the left shows the integration of sin³xcos²x.

Integration of sin²x cos²x

• If both m and n are even, make the substitution the half-angle identities
  
        sin² x = 1/2(1 - cos2x)
        cos² x = 1/2(1 + cos2x)
  
  and substitute u = 2x.
  
  
  The diagram on the left shows the integration of sin²xcos²x.

Integration of tan³x sec⁴x

The second type of combinations of trigonometric functions is powers of tangent and secant.

Form 2:

The following general strategy is used in evaluating integrals of the form , where are integers.

• If n is even (n = 2k, where k is integer), make the substitution sec²x = 1 + tan²x and substitute u = tan x.
  
  
  The diagram on the left shows the integration of tan³xsec⁴x.

• If m is odd (m = 2k +1, where k is integer), make the substitution tan²x = sec²x - 1 and substitute u = sec x.
  
  
  The diagram on the left shows the integration of tan³xsec³x.
  

The third type of combinations of trigonometric functions is production of sine and cosine.

Form 3: and

The following identities are used to evaluate the integrals have form 3.

• sin mx cos nx = 1/2(sin(m - n)x + sin(m + n)x)
• sin mx sin nx = 1/2(cos(m - n)x - cos(m + n)x)
• cos mx cos nx = 1/2(cos(m - n)x + cos(m + n)x)
  The diagram on the left shows the integration of sin2xcos3x.

When an integral of the form arises, where a is a constant, the substitution u = a² + x² is difficult to use because of the root sign. If changing the variable x to θ by the substitution x = asinθ, then using the identity 1 - sin²θ = cos²θ, the root sign can be get rid of. This method is called trigonometric substitutions. There are three principal type of trigonometric substitutions.

Type 1: If occurs in an integrand, then substitute x = a tanθ, θ lie in (-π/2, π/2).

Type 2: If occurs in an integrand, then substitute x = a sinθ, θ lie in [-π/2, π/2].

Type 3: If occurs in an integrand, then substitute x = a secθ, θ lie in [0, π/2) or [-π, 3π/2).