Trigonometric integrals
Integrals of form ∫ R ( sinx, cosx) dx
Here R is THE rational function of sin x and cos x.This can be translated in integrals of the rational function by substitution: tan(x/2) = t. This is the called as universal substitution. In this case
.
At times, instead of the substitution tanx/2 = t, it is more beneficial to make the substitution cot x/2 = t
Universal substitution leads to very cumbersome calculations. Which is indicated below are those cases where the aim can be achieved with help of simpler substitutions.
(a) If R(-sin x, cos x) = -R(sin x, cos x), substitute cos x = t
(b) If R(sin x, -cos x) = -R(sin x, cos x), substitute sin x = t
(c) If R(-sin x, - cos x) = R(sin x, cos x), substitute tan x = t
Integrals of the form:
Rule for (i) : In this integral express numerator as l (Denominator) + m(d.c. of denominator) + n. Find l, m, n by comparing the coefficients of sinx, cosx and constant term and split integral into sum of 3 integrals.
Rule (ii) : Express numerator as l (denominator) + m(d.c. of denominator) and find out l and m shown above
Example: Evaluate 
.
Solution: If in expression 
we substitute -sinx for sin x, then integrand will change its sign. Thus, we take advantage of substitution
t = cosx; dt = - sinx dx. This gives
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