Formula Sheet for Calculus 2 (201-NYB-05)
Trigonometry
TEST #1
Exponential/Log Functions
Integrals
eln(x) = x
Z
e0 = 1,
ln(ex ) = x
ln(1) = 0,
lim ex = 0
x→−∞
b
f (x)dx = F (b) − F (a)
ln(e) = 1
a
Z
xn dx =
lim ln(x) = −∞
x→0+
Z
1
dx = ln |x| + C
x
Z
ex dx = ex + C
ln(xy) = ln(x) + ln(y)
ln(x/y) = ln(x) − ln(y)
xn+1
+C
n+1
y
ln(x ) = y ln(x)
for n 6= 1
Z
sin(x)dx = − cos(x) + C
Derivatives
Z
d
dx
d
dx
d
dx
d
dx
d
dx
d
dx
opp
hyp
hyp
csc(θ) =
opp
opp
tan(θ) =
adj
sin(θ) =
1
sin(θ)
sin(θ)
tan(θ) =
cos(θ)
csc(θ) =
adj
hyp
hyp
sec(θ) =
adj
adj
cot(θ) =
opp
cos(θ) =
1
cos(θ)
cos(θ)
cot(θ) =
sin(θ)
sec(θ) =
sin2 (θ) + cos2 (θ) = 1
1 + tan2 (θ) = sec2 (θ)
1 + cot2 (θ) = csc2 (θ)
sin(2θ) = 2 sin(θ) cos(θ)
cos(2θ) = cos2 (θ) − sin2 (θ)
1 − cos(2θ)
sin2 (θ) =
2
1 + cos(2θ)
2
cos (θ) =
2
Inverse Trig Functions
arcsin(x)
⇒ range = [−π/2, π/2]
arccos(x)
⇒ range = [0, π]
arctan(x)
⇒ range = (−π/2, π/2)
cos(x)dx = sin(x) + C
(c) = 0
Z
sec2 (x)dx = tan(x) + C
Z
csc2 (x)dx = − cot(x) + C
(xn ) = nxn−1
(f (x) ± g(x)) = f 0 (x) ± g 0 (x)
Z
(f (x)g(x)) = f 0 (x)g(x) + f (x)g 0 (x)
f (x)
f 0 (x)g(x) − f (x)g 0 (x)
=
g(x)
g(x)2
Z
(f (g(x))) = f 0 (g(x)) · g 0 (x)
Z
d x
(e ) = ex
dx
d
1
(ln(x)) =
dx
x
sec(x) tan(x)dx = sec(x) + C
csc(x) cot(x)dx = − csc(x) + C
tan(x)dx = − ln | cos(x)| + C
Z
cot(x)dx = ln | sin(x)| + C
Z
sec(x)dx = ln | sec(x) + tan(x)| + C
Z
d
dx
d
dx
d
dx
d
dx
d
dx
d
dx
d
dx
d
dx
d
dx
d
dx
d
dx
d
dx
csc(x)dx = − ln | csc(x) + cot(x)| + C
(sin(x)) = cos(x)
(cos(x)) = − sin(x)
Z
1
1
−1 x
dx
=
tan
+C
x2 + a2
a
a
(tan(x)) = sec2 (x)
Integration by parts
(sec(x)) = sec(x) tan(x)
R
(cot(x)) = − csc2 (x)
u dv = uv −
R
v du
(csc(x)) = − csc(x) cot(x)
Trig substitution
1
1 − x2
1
(arccos(x)) = − √
1 − x2
1
(arctan(x)) =
1 + x2
1
(arccsc(x)) = − √
x x2 − 1
1
(arcsec(x)) = √
x x2 − 1
1
(arccot(x)) = −
1 + x2
a2 − x2
(arcsin(x)) = √
2
⇒
x = a sin θ
2
x −a
⇒
x = a sec θ
x2 + a2
⇒
x = a tan θ
Partial fractions (examples)
·
=
(x + 1)(3x − 4)
·
A
=
+
x(x + 2)2
x
·
A
=
+
x(x2 + 3)
x
A
B
+
x+1
3x − 4
B
C
+
x+2
(x + 2)2
Bx + C
x2 + 3