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.

~ 146. Fundamental theorems concerning the definite integral.

251

VI. Integration by parts. (Partial integration.)
Let the functions 'p(x), f(x), as also their product be integrable;
further let p (x) be a function everywhere continuous and have the
integrable derived function (p'(x), so that therefore (see ~ 147):
x
j 'p'(y)dy = Yp() - (c),
where c and x mean arbitrary points in the interval of integration;
we have then:
b b x
Jf(x)p (x) xdx=jf(x) J'(y)dy + (p(c) dx
a a c
b b x
- p(C)f(x) dx + cjx f'(x)j'(y) zdy
a a c
Now putting c= a, we find:
b b b x
I) j(x) p (x) d x ==p (ca)jf (x) dx + d x gf(x)J (Y) (dy-}
a a a a
a b
Putting c=b, and J p'(y)dy = j'(y)dy, we find:
b x
b b b b
II) Jf (x) p (x) c x (b) j f (x) - cl x d f (x )j' (y) cdy.
a a a x
It will be proved in ~ 168 that the order of these integrations
on the right can be interchanged, at least, if f(x) and (p'(y) remain
finite within the domain of integration; that in fact we have:
b x b b
jdcx Af(x)J '(y)dy = r cy (y) ffx) clxi
a a a y
b l b
j dx f(x)J p'(y)dy f - j d \' (y)jf(x) dx\
a x a ac
Accordingly we obtain from I) and II) the formulas:
b b b b
j;(x)(x) x_ g(ca) (x) d + Jdy \p' {(y) jf(x) dxj,
a( a a y
b b b y
j f(x) 'p (x) d x - 'p (b)J (x) dx - j dy (p(y)J (x) d x.
a a a a