Integration of Basic Irrational Functions

Integrals class 12 of Class 12

Integration of Trigonometric Functions

(a) To evaluate integrals of the form I = Integration of Trigonometric Functions where m and n are rational numbers.

(1) Substitute sin x = t if n is odd +ve integer

(2) cos x = t if m is odd +ve integer

(3) tan x = t if m+n is a –ve even integer

(b) Integrals of the form

Integration of Trigonometric Functionswhere R is a rational function of sin x and cos x are transformed into integers of a rational function by the substitution.

Integration of Trigonometric Functions where -π < x < π

we then have

Integration of Trigonometric Functions

Integration of Trigonometric Functions

Certain functions may be extremely complex rational functions, but in most of the cases the integers can be simplified by the following substitution.

Substitute Integral of the form

(a) sin x = t Integration of Trigonometric FunctionsR (sin x) . cos x dx

(b) cos x = t Integration of Trigonometric FunctionsR (cos x) sin x dx

(c) tan x = t If the integral is dependent only on tan x

(d) tan x = t If the integrand has the form R (sin x cos x) but sin x and cos x appear only in even power.

(e) cos x = t If R(-sin x, cos x) = -R (sin x, cos x)

(f) sin x = t If R (sin x, - cos x) = - R (sin x, cos x)

Integration of Rational Functions

Every rational function can be represented in the form P(x) / Q(x) where P(x) and Q(x) are polynomials  i.e., Integration of Trigonometric Functions assuming of course that the polynomials do not have any common root.

If the fraction is improper, then we can always write Integration of Trigonometric Functions

Just as Integration of Trigonometric Functions

Few Cases

(a) Integration of Trigonometric Functions

(b) Integration of Trigonometric Functions

(c) Integration of Trigonometric Functions

(d) Integration of Trigonometric Functions

 

Integration of Trigonometric Functions

(e) If 4ac – b2 < 0 then (c) fails and we can reorganize.

Integration of Trigonometric Functions

(f) If the integrand is not any of the above forms we decompose the expression into partial fractions and integral separately. For example, I = Integration of Trigonometric Functions

To start we change the expression to an algebraic one by putting tan x = t

we get   Integration of Trigonometric Functions

= ln|1 + t| = ln |1 + tanx| + c

 
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