Calculus: Analytic Geometry and Calculus, with Vectors

8.2 Trigonometric integrands 449

foundation of human experience and that things not closely related to polynomials

are more ethereal. This matter can be of interest to us now because,even in

quite elementary mathematics, the trigonometric functions are called trans-

cendental functions. For example, the assertion that sinx is transcendental

means that there do not exist polynomials Qo, Ql, , Q. in x such that Q"

is nontrivial and (4) is true for each x when f(x) = sin x. Let us now think

briefly about real numbers. A number x is said to be an algebraic number if there

exist integers ko, k1, k2,. , k" not all zero such that

ko+kix+k2x2-{- ... +kx"=0.

Thus a number x is algebraic if it is a zero of a nontrivial polynomial in x having
integer coefficients. A number x which is not an algebraic number is said to be
a transcendental number. The numbers it and e are transcendental. Nobody
can prove these things unless he devotes very much time and energy to the opera-
tion, but nobody requires us to be so busy digging ditches that we never look at
the stars. We can know that there exist theories of algebraic functions and
algebraic numbers and that these theories invite the attention of persons who
have completed studies of analytic geometry and calculus.

8.2 Trigonometric integrands Before introducing chain extensions

and other modifications of the formulas, we systematically work out

formulas for integrals of the six trigonometric functions. Even though

the last three or four have minor importance, we must learn about them

to be respectable. The first two are

(8.21) f sin x dx = - cos x + c, f cos x dx = sin x + c.

They are immediate consequences of the formulas for derivatives of

sines and cosines, since the formula

ff(x) dx = F(x) + c

is valid over an interval if and only if F'(x) = f(x) over the interval.

It is necessary to keep negative signs in their places when we differentiate

and integrate sines and cosines, and it happens that a foolish little trick

enables us to permanently remember how the signs go. We can men-

tally write

(8.22) 5 derivative

integral

sine cosine

in their natural orders and remember that "like things give plus," so

differentiating sines and integrating cosines give plus signs, but that

"unlike things give minus," so differentiating cosines and integrating

sines give minus signs.

If an interval contains one of the points nir ± 7r/2, there can be no F

such that f tan x dx = F(x) + c over that interval. When x is confined