Интеграл: формулы интегралов и правила вычисления

c ÌàòÁþðî ðåøåíèå çàäà÷ ïî âûñøåé ìàòåìàòèêå, òåîðèè âåðîÿòíîñòåé
Íåîïðåäåëåííûé èíòåãðàë
Ïðàâèëà èíòåãðèðîâàíèÿ
Z
Z
0
f (x) dx = f (x) + C,
Z
C · f (x) dx = C ·
Z
(f (x) ± g(x)) dx =
Z
Åñëè
Z
f (x) dx = F (x) + C, òî
Z
f (x) dx ±
Z
f (x) dx,
g(x) dx,
1
f (ax + b) dx = F (ax + b) + C.
a
Òàáëèöà èíòåãðàëîâ
Z
Z
xn+1
x dx =
+ C.
n+1
n
Z
ax
a dx =
+ C, a > 0, a 6= 0.
ln a
Z
x
Z
ex dx = ex + C.
Z
cos x dx = sin x + C.
sin x dx = − cos x + C.
Z
Z
tg x dx = − ln | cos x| + C.
Z
Z
ctg x dx = ln | sin x| + C.
Z
1
dx = tg x + C.
cos2 x
1
1
x
dx
=
arctg
+C
a2 + x2
a
a
Z
√
1
x
− arcctg + C .
a
a
1
x
x
dx = arcsin + C − arccos + C .
a
a
a2 − x2
Z
sh x dx = ch x + C.
Z
dx
= ln |x| + C, x 6= 0.
x
1
dx = thx + C.
ch2 x
1
dx = − ctg x + C.
sin2 x
Z
a − x
1
1
+ C.
dx =
ln x2 − a2
2a a + x Z
√
1
√
dx = ln |x + x2 ± a2 | + C.
x2 ± a2
Z
ch x dx = sh x + C.
Z
1
dx = −cthx + C.
sh2 x