Integrals of the rational functions containing sine and cosine

Integral calculus

The indefinite integral

Trigonometric integrals

Integrals of the rational functions containing sine and cosine,

∫ R (sin x, cos x) dx

Trigonometric integrals of the form

Integrals of rational functions containing sine and cosine, ∫ R (sin x, cos x) dx

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. Therefore,

Example: 101. Evaluate

Solution:

Example: 102. Evaluate

By substituting,

Solution:

The same integrals can be solved using another method, see examples 17. and 18.

Example: 103. Evaluate

By substituting,

Solution:

Example: 104. Evaluate

By substituting,

Solution:

Calculus contents F