Formula Sheet
R
R
(1) Integration By Parts: u(x)v 0 (x)dx = u(x)v(x) − u0 (x)v(x)dx.
(2) Partial Fractions Integral: If c 6= d then
Z
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 00 + by 0 + cy = 0.
• If b2 − 4c > 0, then r1 and r2 are two distinct solutions of the characteristic
equation and
y = C1 er1 t + C2 er2 t ,
where C1 and C2 are constants.
• If b2 − 4c = 0, then there is only one solution of the characteristic equation,
r = −b/2, and
y = C1 tert + C2 ert .
• If b2 − 4c < 0, then the solutions of the characteristic equation are of the form
r = α ± βi and
y = C1 eαt cos(βt) + C2 eαt sin(βt).
(4) For linear non-homogeneous d.e. with constant coefficients:
If f (x) is
then try yp (x) in the form of
polynomial
polynomial of same degree
n
n−1
an x + an−1 x
+ · · · + a0 An xn + An−1 xn−1 + · · · + A0
bekx
Bekx
b sin(ax) or b cos(ax)
B sin(ax) + C cos(ax)
(5) Variation of Parameters for Second Order Equations: If y1 and y2 are linearly independent solutions of the equation y 00 + p(t)y 0 + q(t)y = 0, then yp =
v1 y1 + v2 y2 is a particular solution of the equation p(t)y 00 + q(t)y 0 + r(t)y = f (t),
where v1 and v2 satisfy the VOP equations
v10 y1 + v20 y2 = 0
f (t)
v10 y10 + v20 y20 =
.
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 0 = A~x, with corresponding
eigenvector ~v = p~ + i~q, then two solutions of the system are:
~x1 = eαt [cos(βt) · p~ − sin(βt) · ~q]
~x2 = eαt [sin(βt) · p~ + cos(βt) · ~q]
1
2
(7) Euler Identity: eiθ = cos(θ) + i sin(θ)
(8) Variation of Parameters for First Order Linear Systems: If ~x1 , ~x2 , . . . , ~xn
are linearly independent vector solutions of the n-dimensional homogeneous linear
system ~x 0 = A~x, then
Z
~xp = Y
Y −1 f~dt
or
Z
~xp = Φ
Φ−1 f~dt
is a particular solution of the system ~x 0 = A~x + f~(t) where Y is the n × n matrix
~x1 ~x2 . . . ~xn .
(9) Numerical Approximation by Euler’s Method:
dy
Given the initial value problem
= f (x, y), y(x0 ) = y0 , Euler’s method with step
dx
size h consists of applying the iterative formula
yn+1 = yn + hf (xn , yn ),
(n > 0)
(10) Differentiation formulas
d n
(x ) = nxn−1
dx
d
1
(ln x) =
dx
x
d x
(e ) = ex
dx
d
(sin(x)) = cos x
dx
d x
(a ) = (ln a)ax
dx
d
(cos(x)) = − sin x
dx
d
(tan(x)) = sec2 x
dx
d
(cot(x)) = − csc2 x
dx
d
(sec(x)) = sec x tan x
dx
d
(csc(x)) = − csc x cot x
dx
d
1
(arcsin(x)) = √
dx
1 − x2
d
−1
(arccos(x)) = √
dx
1 − x2
d
1
(arctan(x)) =
dx
1 + x2
d
(sinh(x)) = cosh(x)
dx
d
(cosh(x)) = sinh(x)
dx
d
1
(tanh(x)) =
dx
cosh2 (x)
3
Here a, b, c, d are constants.
A Short Table of Indefinite Integrals
I. Basic Functions
Z
1
xn+1 + C, (n 6= −1)
1.
xn dx =
n+1
Z
1
2.
dx = ln |x| + C
Z x
ax dx =
3.
1 x
a
ln a
+C
Z
4.
Z
1
sin ax dx = − cos ax + C
a
Z
1
6.
cos ax dx = sin ax + C
a
Z
1
7.
tan ax dx = − ln | cos ax| + C
a
5.
ln x dx = x ln x − x + C
II.ZProducts of ex , cos x, and sin x
1
8.
eax sin(bx) dx = 2
eax [a sin(bx) − b cos(bx)] + C
2
a
+
b
Z
1
9.
eax cos(bx) dx = 2
eax [a cos(bx) + b sin(bx)] + C
2
a
+
b
Z
1
[a cos(ax) sin(bx) − b sin(ax) cos(bx)] + C, a 6=
10.
sin(ax) sin(bx) dx = 2
b − a2
b
Z
1
11.
cos(ax) cos(bx) dx = 2
[b cos(ax) sin(bx) − a sin(ax) cos(bx)] + C, a =
6
b − a2
b
Z
1
12.
sin(ax) cos(bx) dx = 2
[b sin(ax) sin(bx) + a cos(ax) cos(bx)] + C, a 6=
b − a2
b
III. Z
Product of Polynomial p(x) with ln x,ex , cos x, and sin x
1
1
13.
xn ln x dx =
xn+1 ln x −
xn+1 + C, n 6= −1, x > 0
2
n
+
1
(n
+
1)
Z
1
1
1
14.
p(x)eax dx = p(x)eax − 2 p0 (x)eax + 3 p00 (x)eax − · · · + C
a
a
a
(+
Z − + − + − + . . .) (signs alternate)
1
1
1
15.
p(x) sin ax dx = − p(x) cos(ax) + 2 p0 (x) sin(ax) + 3 p00 (x) cos(ax) − · · · + C
a
a
a
(− + + − − + + − − . . .) (signs alternate in pairs)
Z
1
1
1
16.
p(x) cos ax dx = p(x) sin(ax) + 2 p0 (x) cos(ax) − 3 p00 (x) sin(ax) − · · · + C
a
a
a
(+ + − − + + − − . . .) (signs alternate in pairs)
4
IV.
Z
17.
Z
18.
Z
19.
Z
20.
Z
21.
Z
22.
Z
23.
Integer Powers of sin x and cos x Z
1
n−1
sinn x dx = − sinn−1 x cos x +
sinn−2 x dx, n positive
n
n Z
1
n−1
cosn x dx = cosn−1 x sin x +
cosn−2 x dx, n positive
n
n
Z
1
cos x
m−2
1
1
dx = −
+
dx, m 6= 1, m positive
m
m−1
m−2
sin x
m
sin
x
− 1 sin x m − 1
1
1 cos x − 1 dx = ln +C
sin x
2
cos x + 1 Z
1
1
sin x
m−2
1
dx =
+
dx, m 6= 1, m positive
m
m−1
m−2
cos x
m − 1 cos x m − 1
cos
x
1
1 sin x + 1 dx = ln +C
cos x
2
sin x − 1 sinm x cosn x dx :
If n is odd, let w = sin x.
If both m and n are even and non-negative, convert all to sin x or all to cos x
(using sin2 x + cos2 x = 1), and use IV-17 or IV-18.
If m and n are even and one of them is negative, convert to whichever function
is in the denominator and use IV-19 or IV-21.
The case in which both m and n are even and negative is omitted.
V.ZQuadratic in the Denominator
x
1
1
24.
dx
=
arctan
+ C, a 6= 0
2
2
a
a
Z x +a
x
bx + c
b
c
2
2
25.
dx
=
ln
|x
+
a
|
+
arctan
+ C, a 6= 0
2
2
2
a
a
Z x +a
1
1
dx =
(ln |x − a| − ln |x − b|) + C, a 6= b
26.
(a − b)
Z (x − a)(x − b)
cx + d
1
27.
dx =
[(ac + d) ln |x − a| − (bc + d) ln |x − b|] + C, a 6=
(x − a)(x − b)
(a − b)
b
√
√
√
2 + x2 ,
2 − x2 ,
VI.
Integrands
involving
a
a
x 2 − a2 , a > 0
Z
dx
x
√
28.
= arcsin
+C
2
2
a
a −x
Z
√
dx
√
= ln |x + x2 ± a2 | + C
29.
2
2
Z √x ± a
Z
√
1
1
2
2
2
2
2
√
30.
a ± x dx =
x a ±x +a
dx + C
2 ± x2
2
a
Z
Z √
√
1
1
2
2
2
2
2
√
31.
x − a dx =
x x −a −a
dx + C
2
x 2 − a2