TABELLA INTEGRALI

TABELLA INTEGRALI
∫0⋅ dx = c
∫ dx = x + c
∫ k ⋅ f ( x) = k ⋅∫ f ( x)
dx
n+
1
n x
∫ xdx = c,( n 1)
n+
1
+ ≠−
n
'
1
n+
1
∫[ f ( x) ] ⋅ f ( x) dx = [ f ( x)
] + c
n + 1
dx
f '(
x)
∫ dx = x + c
2 x
∫ dx = f ( x)
+ c
2 f ( x)
∫ sinx dx = − cos x + c
∫ sen f ( x) ⋅ f '( x) dx = − cos f ( x)
+ c
∫ cos x dx = sen x + c
∫ cos f ( x) ⋅ f '( x) dx = sen f ( x)
+ c
1 f '( x)
∫ dx = tgx + c
2
dx tg f x c
cos x
∫ 2
= ( ) +
cos f ( x)
1 f '( x)
∫ dx = −ctgx
+ c
dx ctg f x c
2
sen x
∫
( )
2
= − +
sen f ( x)
dx
f '( x)
∫ = arcsinx + c
2
∫
dx = arcsin f ( x)
+ c
2
1−
x
1−
f ( x)
dx
∫ = arctg x + c
2
1+
x
dx
∫ = ln x + c
x
∫
1+
[ ]
f '( x)
[ f ( x)
]
2
dx = arctg f ( x)
+ c
f '( x)
∫
f x dx = ln
( )
f ( x ) + c
x x
∫ e dx = e + c
∫ e f ( x) f '( x)
dx = e f ( x)
+ c
x
x
a
∫ a dx = + c
lna
m+
1
m
( x + a)
∫( x + a)
dx = + c
m + 1
dx 1 x
∫ = arctg + c
2 2
a + x a a
n+
1
n
( a + bx)
∫( a + bx)
dx = + c
b( n + 1)
1 1 1 +
∫ 2
= +
1−
x dx x
ln c
2 1−
x
f ( x)
f ( x)
a
∫ a f '( x)
dx = + c
ln a
n+
1
n
( a + bx)
∫( a + bx)
dx = + c
b( n + 1)
dx 1
∫ 2
= − + c
( a + bx) b( a + bx)
dx 1
∫ 2
= − + c
( a + bx) b( a + bx)
∫
1
= tg x + c
1+
cosx 2

1
∫ = − +
1−
cos x dx ctg x c ∫ tgx dx = − lncos x + c
2
∫ctg xdx = ln sinx + c
dx
∫ = lntg x + c
sinx 2
dx
∫ =
cosx
1
2
1+
ln
1−
sinx
sinx
+ c
2
∫ arccosx dx = xarccosx − 1− x + c
∫
1
2
arcctgxdx = xarcctgx + ln 1+ x + c
2
2
∫arcsinx dx = x arcsinx + 1− x + c
1
2
∫ arctg xdx = x arctg x − ln1+ x + c
2
dx 1
∫ = ln a + bx + c
a + bx b
dx 1 ⎛ b
= ⎜ ⋅
a + bx ab ⎝ a x ⎞
∫ ⎟ +
⎠
c dx
2 arctg
a bx dx 1 ab + bx
∫ 2
= ln + c
− 2 ab ab − bx
2
2 2
x a x
a x dx a
2 x
2
dx
∫ − = − + arcsin + c
2 2 a ∫ = arcsin x + c
2 2
a − x a
2
x
a
2 2 2 2
2 2
2
∫ a + x dx = a + x + ln x + a + x + c
3
2 2 ∫ a + bx dx = ( a + bx)
+ c
3b
dx
2 2
dx 2
∫ = ln x + a ± x + c
2 2
∫ =
a ± x
a + bx b a + bx + c
dx 1 x 1
2
c
x − 1
= −
∫ ln
2 x + 1
+
∫ lnxdx = xlnx − x + c
lnx
ln
x dx x 1
2
1
∫ = − − + c
cos ( cos )
2
x x
∫ xdx = x + sinx x + c
2
∫ sin xdx = ( x − sinx cos x) + c cos ( x − a) dx = ( x + sin( x − a) cos( x − a)
+ c
2
2
dx
tg x
dx ⎛π
x⎞
∫ = ln + c
lntg
c
sinx 2
∫ = − ⎜ − ⎟ +
cosx
⎝ 4 2⎠
2 1
∫
2
1