Remember that you can always check/verify integration problems/formulas by simply differentiating your answer.
Basic Properties/Formulas/Rules
1.
k f (x)dx
∫=kf(x)dx
∫
2.
f(x)±g(x)dx
∫=f(x)dx
∫±g(x)dx
∫
Polynomial/Rational Integrals
3.
0dx
∫=C
4.
k dx
∫=kx +C
5.
xndx
∫=xn+1
n+1
+C,n≠ −1
6.
x−1dx =
∫dx
x
∫=ln x+C
Trig Integrals
7.
sin x dx
∫=−cos x+C
8.
cos x dx
∫=sin x+C
9.
sec2x dx
∫=tan x+C
10.
csc2x dx
∫=−cot x+C
11.
sec xtan x dx
∫=sec x+C
12.
csc xcot x dx
∫=−csc x+C
Exponential Integrals
13.
exdx
∫=ex+C
14.
axdx
∫=ax
ln a
+C
Hyperbolic Integrals
15.
sinh x dx
∫=cosh x+C
16.
cosh x dx
∫=sinh x+C
17.
sech2x dx
∫=tanh x+C
18.
csch2x dx
∫=−coth x+C
19.
sech xtanh x dx
∫=−sech x+C
20.
csch xcoth x dx
∫=−csch x+C
Inverse Trig/Hyperbolic Integrals
21.
dx
a2−x2
∫=sin−1x
a
⎛
⎝
⎜⎞
⎠
⎟+C
22.
dx
x2±a2
∫=ln x+x2±a2
( )
+C
23.
dx
a2+x2
∫=1
a
tan−1x
a
⎛
⎝
⎜⎞
⎠
⎟+C
24.
dx
a2−x2
∫=1
2a
ln a+x
a−x
+C
25.
dx
x x2−a2
∫=1
a
sec−1x
a
⎛
⎝
⎜⎞
⎠
⎟+C
26.
dx
x a2±x2
=
∫−1
a
ln a+a2±x2
x
⎛
⎝
⎜⎞
⎠
⎟+C
Note: The inverse trig integrals (left column) will look more familiar if you let
a=1
. However, letting
a
be an arbitrary
constant we can derive the formulas above, which are a great deal more general (hence much more useful).
Note: The inverse hyperbolic integrals (right column) may not be immediately recognizable because the right-hand sides
are written using the logarithmic definitions of the inverse hyperbolic functions. This is customarily done because it
allows you to write all five integrals using only the three equations above. As with the inverse trig integrals, they will
look more familiar if you let
a=1
(and look at the inverse hyperbolic definitions). Again, letting
a
be arbitrary
yields formulas that are more general.
Math 1920 – Basic Integration Formulas
