MTH 202 REFERENCE
Formulas
Derivative Rules
Constant
Constant Multiple
d
c
dx
=0
d
x
dx
=1
d
cx
dx
=c
Power Rule
d
(xr )
dx
= rxr−1
Exponential
d
(ax )
dx
= ax ln(a)
d
(ex )
dx
= ex
Log
Product Rule
Quotient Rule
Chain Rule
Trig Derivatives
Inverse Trig Derivatives
d
dx
loga x =
d
dx
ln x =
1
x ln(a)
1
x
d
(f (x)g(x))
dx
h
i
= f 0 (x)g(x) + g 0 (x)f (x)
=
f 0 (x)g(x)−g 0 (x)f (x)
(g(x))2
d
[f (g(x))]
dx
= f 0 (g(x))g 0 (x)
d
dx
f (x)
g(x)
d
dx
sin x = cos x
d
dx
cos x = − sin x
d
dx
tan x = sec2 x
d
dx
csc x = − csc x cot x
d
dx
sec x = sec x tan x
d
dx
cot x = − csc2 x
d
dx
arcsin x =
√ 1
1−x2
d
dx
arccos x =
√ −1
1−x2
d
dx
arctan x =
1
1−x2
1
Trig Identities
Pythagorean Identities sin2 x + cos2 x = 1
1 − sin2 x = cos2 x
1 − cos2 x = sin2 x
tan2 x = sec2 x − 1
cot2 x = csc2 x − 1
Sum and Difference Identities sin(a ± b) = sin(a) cos(b) ± cos(a) sin(b)
cos(a ± b) = cos(a) cos(b) ∓ sin(a) sin(b)
Double Angle Identities sin 2x = 2 sin x cos x
cos 2x = cos2 x − sin2 x
sin x cos x = 21 sin 2x
Half Angle Identities sin2 x = 12 (1 − cos 2x)
cos2 x = 21 (1 + cos 2x)
sin a cos b = 12 [sin(a − b) + sin(a + b)]
sin a sin b = 12 [cos(a − b) − cos(a + b)]
cos a cos b = 12 [cos(a − b) + cos(a + b)]
Antiderivative Rules
f and g are continuous functions.
Constant
R
k dx = kx + C
Power Rule
R
xn dx =
R
1
x
R
ax dx =
R
ex dx = ex + C
Exponential
Substitution Rule
Integration by Parts
Rb
a
R
xn +1
n+1
+ C, (n 6= −1)
dx = ln |x| + C
ax
ln a
+C
f (g(t))g 0 (t) dt =
R g(b)
g(a)
f (x) dx, (g 0 (t) 6= 0, t ∈ [a, b])
f (x)g 0 (x) dx = f (x)g(x) −
2
R
f 0 (x)g(x) dx
Trig Antiderivatives
Inverse Trig Antiderivatives
R
sin x dx = − cos x + C
R
cos x dx = sin x + C
R
sec2 x dx = tan x + C
R
csc x cot x dx = − csc x + C
R
sec x tan x dx = sec x + C
R
csc2 x dx = − cot x + C
R
sec x dx = ln | sec x + tan x| + C
R
tan x dx = ln | sec x| + C
R
csc x dx = − ln | csc x + cot x| + C
R
cot x dx = ln | sin x| + C
R
1
x2 +a2
R
dx =
√ 1
a2 −x2
1
a
arctan
dx = arcsin
R√
1 + x2 dx =
1
2
x
a
x
a
+C
+C
√
√
x 1 + x2 + ln | 1 + x2 + x| + C
Other Integration Formulas
Reduction Formulas
R
sinn x dx = − sin
R
cosn x dx =
n−1
x cos x
n
sin x cosn−1 x
n
+
+
n−1
n
n−1
n
R
Volume by disks (around x-axis) V = π (r(x))2 dx
R
Volume by washers (around x-axis) V = π (ro (x))2 − (ri (x))2 dx
R
Volume by shells (around y-axis) V = 2π r(x)f (x) dx
Rbq
dy 2
Arc Length L = a 1 + dx
dx
r
2
Rd
L = c 1 + dx
dy
dy
R
Surface Area A = 2πy ds
q
Rb
dy 2
A = a 2πy 1 + dx
dx
r
2
Rd
A = c 2πy 1 + dx
dy
dy
3
R
R
sinn−2 x dx
cosn−2 x dx
Series
P
xn
Common Series ex = ∞
n=0 n!
P
(−1)n+1 x2n+1
sin x = ∞
n=0
(2n+1)!
P
(−1)n x2n
cos x = ∞
n=0
(2n)!
Harmonic Series.
∞
X
1
diverges.
n
n=1
P Series.
∞
X
1
diverges if p ≤ 1
p
converges if p > 1
n
n=1
Geometric Series.
∞
X
r
n
n=1
diverges if |r| ≥ 1
converges if |r| < 1
P
P
Comparison Test. Suppose
an and
bn are series with positive terms.
P
P
(1) If P bn is convergent and an ≤ bn for all n, thenP an is also convergent.
(2) If
bn is divergent and an ≥ bn for all n, then
an is also divergent.
Limit Comparison Test. Suppose that
P
an and
P
bn are series with positive terms.
If
an
=c
a→∞ bn
where c is a finite number and c > 0, then either both series converge or both series diverge.
lim
Integral Test. Suppose f isPa continuous, positive, decreasing function on [1, ∞) and
∞
let
a
R ∞ n = f (n). Then the series n=1 an is convergent if and only if the improper integral
f (x) dx is convergent.
1
Alternating Series Test. If the alternating series
∞
X
(−1)n−1 bn = b1 − b2 + b3 − b4 + b5 − . . .
bn > 0
n=1
satisfies
(1) bn+1 ≤ bn for all n
(2) limn→∞ bn = 0
then the series is convergent.
Absolute Convergence Test. If
P
|an | converges,
4
P
an converges.
Ratio Test.
∞
X
an+1 (1) If lim = L < 1, then the series
an is absolutely convergent.
n→∞
an n=1
an+1 an+1 = ∞, then the series diverges.
(2) If lim = L > 1 or lim n→∞
n→∞
a
a
n
n an+1 = 1, then the Ratio Test is inconclusive.
(3) If lim n→∞
an Root Test.
∞
X
p
n
|an | = L < 1, then the series
an is absolutely convergent.
(1) If lim
n→∞
n=1
p
p
(2) If lim n |an | = L > 1 or lim n |an | = ∞, then the series diverges.
n→∞ p
n→∞
(3) If lim n |an | = 1, then the Ratio Test is inconclusive.
n→∞
Parametric Equations
Let x = f (t) and y = g(t).
Derivative
dy
dx
=
dy/dt
dx/dt
Rb
g(t)f 0 (t)dt
Rβp
Arc Length L = α (f 0 (t))2 + (g 0 (t))2 dt
Integral
a
Polar Equations
Derivative
dy
dx
Integral
Rb
=
dy/dθ
dx/dθ
1
(r(θ))2
a 2
5
dθ