129 
J = \ logxclx = logX . X 
u . d v u . v 
= x log x — | d x = x log x — x + C = x (log x — 1) -j- C. 
186. Beispiel. J = Jx n . logx . dx. 
Setzen wir log x — u und x 11 dx = d v, so ist 
d x y?1 "t - 2 
du = — und v 
x 
J = \ log x . x“ d x = log x 
i.dv 
X n + 1 
n —1 
log X r-^r { X n . d 5 
n 4-1 J 
X n + 1 
n+I' 
dx 
' n + 1’ 
x 11 + 1 d x 
n+I ■ X 
v . du 
•Ti) + a 
X 11 "f" 1 X n “I - * X n "t" i / 
= ^ logx f- = ( log; 
187. Beispiel. J — j* x . cosxdx. 
Wir setzen x = u, cosxdx = dv; 
d u = d x, v — sin x. 
J = J x . cos x d x = x sin x — f sin x d x = x sin x -)- cos x-f C. 
188. Beispiel. J = Jarctgxdx. 
arc tg x = u, d x = d v, 
, d x 
du= r+^ v = 
J = Care x ■ d x — arc tg x . x — C 
1 r d(l -!- x 2 ) 1 . . , 0x . ~ 
= x. arctgx — y ) —! j' x2 - = x.arctgx — -g-log(l -f x 2 ) + C. 
189. Beispiel. J = \ sin 2 x dx. 
sin x = u, sin x d x = d v, 
d u = cos x d x, v — — cos x. 
J = Jsin x . sinxdx = sin x . (— cosx) — [ (— cosx).cosxdx = 
= — sinx.cosxi| cos 2 xdx =— sinxcosx -|-f(l— sin 2 x) dx = 
= — sin x , cos x -f- j* d x — j sin 2 x . d x = 
= — sin X COS X -j- x — j sin 2 x d X. 
d x 
l'+x 1 
Mandl, Mathematik. 
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