An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 107 —108. Fundamental formulas and rules of integration.

183

b
7) ~ -- cot b +- cot a, (in every interval from a to b,
which does not contain zr or multiples of zr).
b
8) and 9) _= =- sinlb - sin-:a - cos-b +- cos-la,
V1 - X2
(-1 a < +- 1, - 1 b 1).
b
10) and 1 1) j — =tan-lb - tan-" a =- cot-'b + cot- a,
a
( - c < a < -~ - c< b + co).
108. We can also assign the indefinite integral for functions
compounded of the simple ones; for this we require the General Rules
that can be derived by inverting the Rules of Differentiation (~ 26).
If:
a) f(x) - (x) + f2 (x) + f (x) +.. + t (x)
we have:
jf(x)dx =; (x) cdx +J 2 (x) dx + J (x) d x +. ~f (x) lx,
that is, the integral of a sum of functions is equal to the sum of the
integrals of the several summands. This is proved by differentiation.
If:
b) F(x) -= (p(x)(x), then F'(x) == f(x) () (x)'(x) + g(x)9'(x),
therefore inverting:
f P(x) '(x)dx +d- (x)9'(x) dx - 9 (x))(x).
If we write this formula:
s c(x) ' (x)dx == 9 (x) (x) - J (x) V (x) dx
or again:
j(x) d (tP (x)) = q (x) p (x) j (x) d (p (x)),
it shows how to reduce the integral of a function, consisting of two
factors of which one can be integrated, to another integral. This
reduction, called the process of integration by p arts, in many cases
simplifies the problem.
Special theorem: If:
F(x) == a((x), then F' (x) = f(x) -- a ' (x),
therefore:
J a '(x) dx = a (p (x) -= a f (x) dx.