Must Know Derivative and Integral Rules!
Table I: General Rules
Derivative Rule
Integration Rule
Rule
Sum/Difference
Rule R
Sum/Difference
R
R
d
0
0
f
(x)
±
g(x)
dx
=
f
(x)dx
± g(x)dx
f
(x)
±
g(x)
=
f
(x)
±
g
(x)
dx
Constant
Multiple
Rule
Constant
Multiple
R
R Rule
d
0
cf (x) = cf (x)
cf (x)dx = c f (x)dx
dx
Product
Rule
Integration
by Parts
R 0
R
d
0
0
f (x)g(x) = f (x)g(x) + f (x)g (x)
f (x)g(x)dx = f (x)g(x) − f (x)g 0 (x)dx
dx
Rule
(no simple rule corresponds)
h Quotient
i
d
dx
d
dx
f (x)
g(x)
=
f 0 (x)g(x)−f (x)g 0 (x)
[g(x)]2
Chain
Rule
f (g(x)) = f 0 (g(x))g 0 (x)
R
U-Substitution
R
f (g(x))g 0 (x)dx = f (u)du
where u = g(x)
Table II: Rules for Specific Functions
Derivative Rule
Integration Rule
Constant Rule
RConstant Rule
d
[c]
=
0
c dx = cx + C
dx
Power Rule
R p Power1 Rule
d
p
p−1
[x ] = px
x dx = p+1 xp+1 + C
dx
for p 6= −1
R
dx
d
ln |x| = x1
= ln |x| + C
dx
x
d
dx
logb |x| =
d x
e
dx
d x
b
dx
d
dx
d
dx
d
dx
1
x ln b
same as above
R x
e dx = ex + C
= ex
R
= (ln b)bx
R
sin(x) = cos(x)
d
dx
cos(x) = − sin(x)
R
d
dx
tan(x) = sec2 (x)
R
sec(x) = sec(x) tan(x)
csc(x) = − csc(x) cot(x)
d
dx
R
cot(x) = − csc2 (x)
d
dx
arctan(x) =
d
dx
arcsin(x) =
d
dx
R
1
1+x2
√ 1
1−x2
1
arccos(x) = − √1−x
2
bx dx =
1 x
b
ln b
+C
cos(x)dx = sin(x) + C
sin(x)dx = − cos(x) + C
sec2 (x)dx = tan(x) + C
sec(x) tan(x)dx = sec(x) + C
csc(x) cot(x)dx = − csc(x) + C
R
csc2 (x)dx = − cot(x) + C
R dx
= arctan (x) + C
1+x2
R dx
√
= arcsin (x) + C
1−x2
same as above
(NOTE: arccos(x) = π2 − arcsin(x))
1
Table III: Additional Integrals
R
+C
sin2 (x)dx = x2 − sin(2x)
4
R
cos2 (x)dx = x2 + sin(2x)
+C
4
R
tan(x)dx = ln | sec(x)| + C
R
cot(x)dx = ln | sin(x)| + C
R
sec(x)dx = ln | sec(x) + tan(x)| + C
R
csc(x)dx = − ln | csc(x) + cot(x)| + C
Table IV: Useful Integrals (these require Integration-by-Parts)
R
ln(x)dx = x ln(x) − x + C
R
logb (x)dx = ln1b (x ln(x) − x) + C
R
arctan(x)dx = x arctan(x) − 12 ln(1 + x2 ) + C
√
R
arcsin(x)dx = x arcsin(x) + 1 − x2 + C
√
R
arccos(x)dx = x arccos(x) − 1 − x2 + C
2