Integration Rules
Basic Differentiation Rules
The rules for you to note/recall:
• The constant rule
Z
a dx = ax + C
• The polynomial rule
Z
x n dx =
x n +1
1
+C =
x n +1 + C
n+1
n+1
• The scalar multiple rule
Z
• The sum rule
Z
k f ( x ) dx = k
f ( x ) ± g( x ) dx =
Z
Z
f ( x ) dx
f ( x ) dx ±
Z
g( x ) dx
• The reciprocal rule
1
dx = ln | x | + C
x
Z
• The exponential rules
Z
e x dx = e x + C
• The trigonometric rules
Z
Z
Z
cos x dx = sin x + C
sin x dx = − cos x + C
sec2 x dx = tan x + C
Other Integration Rules
• Integration by Substitution
If the function u = g( x ) has a continuous derivative and f is continuous then
Z
f ( g( x )) g0 ( x ) dx =
Z
f (u) du .
Look for one part of the function that is the derivative of the other part.
For example:
Z
(ln x )2 − 4 ln x
dx.
x
Solution:
Let u = ln x , thus
du
1
=
,
dx
x
dx = x du .
Z
(ln x )2 − 4 ln x
(u2 − 4u)
dx =
x du
x
x
Z
=
u2 − 4u du
Z
u3
− 2u2 + C
3
(ln x )3
− 2(ln x )2 + C .
3
=
=
• Integration by Parts
If u and v are functions of x and have a continuous derivative, then
Z
dv
u
dx = uv −
dx
Z
v
du
dx .
dx
Guidelines for integration by parts:
– Try letting u be the portion of the integrand whose derivative is a simpler
function than u.
dv
– Try letting
be a trigonometric function or an exponential function.
dx
For example:
Z
xe x dx.
Solution:
Let
du
=1
dx
u=x →
dv
= ex → v = ex
dx
Integrating by parts gives:
Z
dv
dx
dx
Z
du
= uv − v
dx
Z dx
xe x dx =
Z
u
= xe x −
e x dx
= xe x − e x + C
= e x ( x − 1) + C .