Download x| + C

Inegration Reference Page
Math 12-D. Benedetto
Power Functions
Trigonometric Functions
Exponentials and Logarithms
xn+1
+ C with n 6= −1
n+1
•
Z
xn dx =
•
Z
1
dx = ln |x| + C
x
•
Z
sin x dx = − cos x + C
•
Z
cos x dx = sin x + C
•
Z
tan x dx = − ln | cos x| + C = ln | sec x| + C
•
Z
sec x dx = ln | sec x + tan x| + C
•
Z
sec2 x dx = tan x + C
•
Z
sec x tan x dx = sec x + C
•
Z
ex dx = ex + C
•
Z
ekx dx =
•
Z
1
dx = ln |x| + C
x
•
Z
1
1
dx = ln |ax + b| + C
ax + b
a
•
Z
ln x dx = x ln x − x + C
1 kx
e +C
k
using Integration By Parts?!
*******************************************************
Review other ex and ln x handout
Inverse Trig. Functions
Hyperbolic Functions
1
dx = arcsin x + C = sin−1 x + C
1 − x2
•
Z
√
•
Z
1
dx = arctan x + C = tan−1 x + C
1 + x2
•
Z
1
√
dx = arcsecx + C = sec−1 x + C
x x2 − 1
•
Z
√
•
Z
x
1
1
1
−1 x
+
C
=
+C
dx
=
arctan
tan
a2 + x2
a
a
a
a
x
x
1
+ C = sin−1
+C
dx = arcsin
a
a
a2 − x2
• sinh x =
ex − e−x
2
• cosh x =
ex + e−x
2
• tanh x =
sinh x
ex − e−x
= x
cosh x
e + e−x
•
d
sinh x = cosh x
dx
•
d
cosh x = sinh x
dx
d
tanh x = sech2 x
dx
Z
• sinh x dx = cosh x + C
•
•
Z
cosh x dx = sinh x + C
•
Z
√
•
Z
1
dx = tanh−1 x + C
1 − x2
•
Z
√
1
dx = sinh−1 x + C
2
1+x
1
x2
−1
dx = cosh−1 x + C
Products of Trig. Functions
•
Z
sinm x cosn x dx =??
•
Z
tanm x secn x dx =??
know even/odd power techniques
Integrand Contains
√
Trig. Substitution
√
√
Trigonometric Identities
Substitute
Identity
a2 − x2
x = a sin θ
sin2 θ + cos2 θ = 1
a2 + x2
x = a tan θ
sec2 θ = 1 + tan2 θ
x2 − a2
x = a sec θ
tan2 θ = sec2 θ − 1
• sin2 θ + cos2 θ = 1
• sec2 θ = 1 + tan2 θ
• sin2 θ =
1 − cos(2θ)
2
• cos2 θ =
1 + cos(2θ)
2
• sin(2θ) = 2 sin θ cos θ
**********************************************************
• cosh2 x − sinh2 x = 1
Integration by Parts
•
Z
•
Z
•
Z
u dv = uv −
Z
v du
′
f (x)g (x) dx = f (x)g(x) −
b
a
b
u dv = uv a −
Z
a
b
v du
Z
g(x)f ′ (x) dx