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Integral calculus
  Trigonometric integrals
 Integrals of rational functions containing sine and cosine
Integrals of the form,        ∫ R (sin x, cos x) dx
where R denotes a rational function of sin x and cos x, can be transformed to a rational function of the new variable t, using substitution  tan (x/2) = t. Then,
therefore, by substituting
  In case the integrand expression does not change the sign when both the sine and the cosine functions change the sign, i.e., if 
R (- sin x,  - cos x)    or    R (sin x, cos x)
we can use the substitution tan x = t. Thus,
 Integrals of rational functions containing sine and cosine examples
 Example:  Evaluate
 Solution:
 Example:  Evaluate
 Substituting,
 Solution:
 Example:  Evaluate
 Substituting,
 Solution:
 Example:  Evaluate
 Substituting,
 Solution:
 Integrals of the hyperbolic functions
Integration of the hyperbolic functions is similar to the integration of the trigonometric functions except we use the following hyperbolic identities or formulas.
When we use the hyperbolic functions in substitutions, solutions are transformed using following relations,
sinh x + cosh x = ex    or     x = ln (sinh x + cosh x),   and
 Integrals of the hyperbolic functions examples
 Example:  Evaluate
 Solution:
 Example:  Evaluate
 Solution:
 Example:  Evaluate
 Solution:
 
 
 
 
 
 
 
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