Basic Integration Formula Sheet
Content
Basic Integration Formula Sheet
Basic Integration Formulas 0 du = 0 du = u + C k du = ku + C un+1 u du = n+1
n
Created by MSW http://www.nvcc.edu/home/mwesterhoff
Integration Formulas u √ = arcsin +C a a2 − u2 1 u √ = arctan +C a a a2 + u2 du 1 |u| √ = arcsec +C a a u u2 − a2 du du
Definite (Riemann) Integral
n ||∆→0|| b
Fundamental Theorem of Calculus I (FTC I)
b
f (x) dx = F (b) − F (a) f (x) dx
a
lim
f (ci )∆xi =
i=1 a
Fundamental Theorem of Calculus II (FTC II) d dx d dx
x
Partitions of equal width: ||∆|| = ∆x Partitions of unequal width: ||∆|| = max(∆xi ) Continuity ⇒ Integrability The Converse is NOT True (see example below)
f (t) dt = f (x)
a u
f (t) dt = f (u)u (with chain rule)
a
(for n = 1) Summation Formulas
Mean Value Theorem for Integration
2
cos u du = sin u + C sin u du = −cos u + C sec u du = tan u + C sec u tan u du = sec u + C csc2 u du = −cot u + C csc u cot u du = −csc u + C eu du = eu + C au du = 1 ln a au + C, a > 0
2
n
Example: = a1 + a2 + . . . + an
0
x dx = 1
b
f (x) dx = f (c)(b − a), where c ∈ [a, b]
a
i=1 n
Area of a Region c = cn Area of the region bounded by the graph of f , x-axis, and x = a and x = b
b
Average Value of a Function on [a, b] f (c) = 1 b−a
b
i=1 n
f (x) dx
a
n(n + 1) i= 2 i=1
n
i2 =
i=1 n
n(n + 1)(2n + 1) 6
Area =
a
f (x) dx
Substitution Method for Integration Let u = g(x), then
b
n2 (n + 1)2 i3 = 4 i=1
n
Properties of Integration
a
du = g (x) ⇒ du = g (x)dx, dx
b g(b)
i4 =
i=1
n(2n + 1)(n +
1)(3n2 30
+ 3n − 1)
1.
a a
f (x) dx = 0
b
f (g(x))g (x) dx ⇒
a a
f (u) du ⇒
g(a)
f (u) du
2.
b
f (x) dx = −
a b c
f (x) dx
b
Numerical Integration Note: Many integrals do not have closed form solutions, so we must use numerical techniques.
n
Area of a Region in a Plane 3.
n
f (x) dx =
a a b
f (x) +
c b
f (x) dx
1 du = ln |u| + C u tan u du = −ln|cos x| + C cot u du = ln|sin x| + C sec u du = ln|sec x + tan x| + C csc u du = −ln|csc x + cot x| + C
I.R. = lim
n→∞
f (mi )∆x
i=1 n
4.
a
kf (x) dx = k
a
f (x) dx
Midpoint Rule: f (x) dx ≈
i=1 b
f
xi + xi−1 2
n
∆x
C.R. = lim
n→∞
f (Mi )∆x
i=1 n
5.
[f (x)dx ± g(x) ± · · · ± k(x)] dx =
b b b
Trapezoidal Rule:
a
f (x) dx ≈
i=1
f (xi ) + f (xi−1 ) 2
∆x
Area = lim
n→∞
f (ri )∆x
i=1
f (x) ±
a b a
g(x) dx ± · · · ±
a
k(x) dx E≤
where ri ∈ [xi−1 , xi ] Left Endpoints: a + b(i − 1)∆x, for i = 1, . . . , n Right Endpoints: a + bi∆x, for i = 1, . . . , n ∆x = (b − a)/n
(b − a)3 [max |f (x)|], for x ∈ [a, b] 12n2
6. 0 ≤
a
f (x) dx for f non-negative
b b
7. If f (x) ≤ g(x) ⇒
a
f (x) dx ≤
a b
f (x) dx b−a [f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) + · · · + 4f (xn−1 ) + f (xn )] 3n
Simpson’s Rule:
a
f (x) dx ≈
E≤
(b − a)5 [max |f (4) (x)|], for x ∈ [a, b] 180n4
Basic Integration Formulas 0 du = 0 du = u + C k du = ku + C un+1 u du = n+1
n
Created by MSW http://www.nvcc.edu/home/mwesterhoff
Integration Formulas u √ = arcsin +C a a2 − u2 1 u √ = arctan +C a a a2 + u2 du 1 |u| √ = arcsec +C a a u u2 − a2 du du
Definite (Riemann) Integral
n ||∆→0|| b
Fundamental Theorem of Calculus I (FTC I)
b
f (x) dx = F (b) − F (a) f (x) dx
a
lim
f (ci )∆xi =
i=1 a
Fundamental Theorem of Calculus II (FTC II) d dx d dx
x
Partitions of equal width: ||∆|| = ∆x Partitions of unequal width: ||∆|| = max(∆xi ) Continuity ⇒ Integrability The Converse is NOT True (see example below)
f (t) dt = f (x)
a u
f (t) dt = f (u)u (with chain rule)
a
(for n = 1) Summation Formulas
Mean Value Theorem for Integration
2
cos u du = sin u + C sin u du = −cos u + C sec u du = tan u + C sec u tan u du = sec u + C csc2 u du = −cot u + C csc u cot u du = −csc u + C eu du = eu + C au du = 1 ln a au + C, a > 0
2
n
Example: = a1 + a2 + . . . + an
0
x dx = 1
b
f (x) dx = f (c)(b − a), where c ∈ [a, b]
a
i=1 n
Area of a Region c = cn Area of the region bounded by the graph of f , x-axis, and x = a and x = b
b
Average Value of a Function on [a, b] f (c) = 1 b−a
b
i=1 n
f (x) dx
a
n(n + 1) i= 2 i=1
n
i2 =
i=1 n
n(n + 1)(2n + 1) 6
Area =
a
f (x) dx
Substitution Method for Integration Let u = g(x), then
b
n2 (n + 1)2 i3 = 4 i=1
n
Properties of Integration
a
du = g (x) ⇒ du = g (x)dx, dx
b g(b)
i4 =
i=1
n(2n + 1)(n +
1)(3n2 30
+ 3n − 1)
1.
a a
f (x) dx = 0
b
f (g(x))g (x) dx ⇒
a a
f (u) du ⇒
g(a)
f (u) du
2.
b
f (x) dx = −
a b c
f (x) dx
b
Numerical Integration Note: Many integrals do not have closed form solutions, so we must use numerical techniques.
n
Area of a Region in a Plane 3.
n
f (x) dx =
a a b
f (x) +
c b
f (x) dx
1 du = ln |u| + C u tan u du = −ln|cos x| + C cot u du = ln|sin x| + C sec u du = ln|sec x + tan x| + C csc u du = −ln|csc x + cot x| + C
I.R. = lim
n→∞
f (mi )∆x
i=1 n
4.
a
kf (x) dx = k
a
f (x) dx
Midpoint Rule: f (x) dx ≈
i=1 b
f
xi + xi−1 2
n
∆x
C.R. = lim
n→∞
f (Mi )∆x
i=1 n
5.
[f (x)dx ± g(x) ± · · · ± k(x)] dx =
b b b
Trapezoidal Rule:
a
f (x) dx ≈
i=1
f (xi ) + f (xi−1 ) 2
∆x
Area = lim
n→∞
f (ri )∆x
i=1
f (x) ±
a b a
g(x) dx ± · · · ±
a
k(x) dx E≤
where ri ∈ [xi−1 , xi ] Left Endpoints: a + b(i − 1)∆x, for i = 1, . . . , n Right Endpoints: a + bi∆x, for i = 1, . . . , n ∆x = (b − a)/n
(b − a)3 [max |f (x)|], for x ∈ [a, b] 12n2
6. 0 ≤
a
f (x) dx for f non-negative
b b
7. If f (x) ≤ g(x) ⇒
a
f (x) dx ≤
a b
f (x) dx b−a [f (x0 ) + 4f (x1 ) + 2f (x2 ) + 4f (x3 ) + · · · + 4f (xn−1 ) + f (xn )] 3n
Simpson’s Rule:
a
f (x) dx ≈
E≤
(b − a)5 [max |f (4) (x)|], for x ∈ [a, b] 180n4
