f'(x)
∫x
a
dx
x
a+1
/(a+1) + C ;a≠-1
f(x)
f'(x)
∫x
a
dx
x
a+1
/(a+1) + C ;a≠-1
C
0
∫1/x dx
ln|x| + C
C
0
∫1/x dx
ln|x| + C
x
a
ax
a-1
∫a
x
dx
a
x
/lna + C ;a>0, a≠-1
x
a
ax
a-1
∫a
x
dx
a
x
/lna + C ;a>0, a≠-1
e
x
e
x
∫e
x
dx
e
x
+ C
e
x
e
x
∫e
x
dx
e
x
+ C
a
x
a
x
lna
∫sinx dx
cosx + C
a
x
a
x
lna
∫sinx dx
cosx + C
ln|x|
1/x
∫cosx dx
-sinx + C
ln|x|
1/x
∫cosx dx
-sinx + C
log
a
|x|
1/(xlna)
∫1/sin
2
x dx
-ctgx + C
log
a
|x|
1/(xlna)
∫1/sin
2
x dx
-ctgx + C
sinx
cosx
∫1/cos
2
x dx
tgx + C
sinx
cosx
∫1/cos
2
x dx
tgx + C
∫1/√(1-x
2
) dx
cosx
-sinx
arcsinx + C
cosx
-sinx
∫1/√(1-x
2
)
dx
arcsinx + C
1/cos
2
x
∫1/(1+x
2
) dx
tgx
arctgx + C
ctgx
-1/sin
2
x
∫f'/f dx= ln|f| + C ;f≠0
tgx
1/cos
2
x
∫1/(1+x
2
) dx
arctgx + C
arcsinx
1/√(1-x
2
)
∫f
n
*f' dx= f
n+1
/(n+1) +C
ctgx
-1/sin
2
x
∫f'/f dx= ln|f| + C ;f≠0
arccosx
-1/√(1-x
2
)
∫f'/√f dx=2√f +C
arcsinx
1/√(1-x
2
)
∫f
n
*f' dx= f
n+1
/(n+1) +C
arctgx
1/(1+x
2
)
∫f'/f
2
dx=-1/f +c
arccosx
-1/√(1-x
2
)
∫f'/√f dx=2√f +C
arcctgx
-1/(1+x
2
)
-sin,cos → cos=t -sinxdx=dt
arctgx
1/(1+x
2
)
∫f'/f
2
dx=-1/f +c
f'(x)=lim
∆x->0
(f(x+∆x)-f(x))/∆x
sin,-cos → sin=t cosxdx=dt
arcctgx
-1/(1+x
2
)
-sin,cos → cos=t -sinxdx=dt
(f ±g)'=f'±g'
-sin-cos →tg=t x=arctg dx=1/(1+t
2
)dt
sin=t/√(1+t
2
) cos=1/√(1+t
2
)
f'(x)=lim
∆x->0
(f(x+∆x)-f(x))/∆x
sin,-cos → sin=t cosxdx=dt
(f ±g)'=f'±g'
-sin-cos →tg=t x=arctg dx=1/(1+t
2
)dt
sin=t/√(1+t
2
) cos=1/√(1+t
2
)
(f*g)'=f'g+fg'
u tg(x/2)=t x/2=arctgt dx=2/(1+t
2
) dt
sin=2t/(1+t
2
) cos=(1-t
2
)/(1+t
2
)
(f*g)'=f'g+fg'
u tg(x/2)=t x/2=arctgt dx=2/(1+t
2
) dt
sin=2t/(1+t
2
) cos=(1-t
2
)/(1+t
2
)
(f/g)'=(f'g-fg')/g
2
∫uv' dx= uv - ∫vu' dx
(f g)=f'(g)*g'
f
g
=f
g
*(g'lnf+g*(f'/f)) f(x
0
+∆x)≈f(x
0
)+f'(x
0
)*∆x
st. y=f(x) w P
0
( x
0
,y
0
): y-y
0
=f'(x
0
)(x- x
0
), y
0
=f(x
0
)
no. y=f(x) w P
0
( x
0
,y
0
): y-y
0
=(-1/f'(x
0
))(x- x
0
), y
0
=f(x
0
)
aul a=lim
x→-∞
f(x)/x b=lim
x→-∞
[f(x)-ax]
aup a=lim
x→∞
f(x)/x b=lim
x→∞
[f(x)-ax]
(f/g)'=(f'g-fg')/g
2
∫uv' dx= uv - ∫vu' dx
(f g)=f'(g)*g'
f
g
=f
g
*(g'lnf+g*(f'/f)) f(x
0
+∆x)≈f(x
0
)+f'(x
0
)*∆x
st. y=f(x) w P
0
( x
0
,y
0
): y-y
0
=f'(x
0
)(x- x
0
), y
0
=f(x
0
)
no. y=f(x) w P
0
( x
0
,y
0
): y-y
0
=(-1/f'(x
0
))(x- x
0
), y
0
=f(x
0
)
aul a=lim
x→-∞
f(x)/x b=lim
x→-∞
[f(x)-ax]
aup a=lim
x→∞
f(x)/x b=lim
x→∞
[f(x)-ax]