Integral Formula Sheet
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1
2
3
4Formula Sheet
(1) Integration By Parts:
u(x)v (x)dx = u(x)v(x)
u (x)v(x)dx.
(2) Partial Fractions Integral: If c = d then
ax + b
1
dx =
(ac + b) ln x
c
(ad + b) ln x
d + K.
(x
c)(x
d)
c
d
(3) For linear homogeneous d.e. with constant coefficients: y + by + cy = 0.
2
If b
4c > 0, then r
and r
are two distinct solutions of the characteristic
1
2
equation and
r
t
r
t
y = C
e
1
+ C
e
2
,
1
2
where C
and C
are constants.
1
2
2
If b
4c = 0, then there is only one solution of the characteristic equation,
r =
b/2, and
rt
rt
y = C
te
+ C
e
.
1
2
2
If b
4c < 0, then the solutions of the characteristic equation are of the form
r = α
βi and
αt
αt
y = C
e
cos(βt) + C
e
sin(βt).
1
2
(4) For linear non-homogeneous d.e. with constant coefficients:
If f (x) is
then try y
(x) in the form of
p
polynomial
polynomial of same degree
n
n 1
n
n 1
a
x
+ a
x
+
+ a
A
x
+ A
x
+
+ A
n
n 1
0
n
n 1
0
kx
kx
be
Be
b sin(ax) or b cos(ax)
B sin(ax) + C cos(ax)
(5) Variation of Parameters for Second Order Equations: If y
and y
are lin-
1
2
early independent solutions of the equation y + p(t)y + q(t)y = 0, then y
=
p
v
y
+ v
y
is a particular solution of the equation p(t)y + q(t)y + r(t)y = f (t),
1
1
2
2
where v
and v
satisfy the VOP equations
1
2
v
y
+ v
y
= 0
1
2
1
2
f (t)
v
y
+ v
y
=
.
1
1
2
2
p(t)
(6) First Order Linear Systems with Complex Eigenvalues: If λ = α + βi
is an eigenvalue of A for the system of equations, x = Ax, with corresponding
eigenvector v = p + iq, then two solutions of the system are:
αt
x
= e
[cos(βt) p
sin(βt) q]
1
αt
x
= e
[sin(βt) p + cos(βt) q]
2
1
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