Basic Integration Formula Sheet

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 du = arctan +C a2 + u2 a a 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
u 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 in a Plane c = cn Area of the region bounded by the graph of f , x-axis, and x = a and x = b
n 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

Substitution Method for Integration f (x) dx Let u = g (x), then
b

i2 =
i=1 n

n(n + 1)(2n + 1) 6

Area = lim

n→∞

f (ri )∆x =
i=1 a

n2 (n + 1)2 i3 = 4 i=1
n

where ri ∈ [xi−1 , xi ] Properties of Integration

du = g (x) ⇒ du = g (x)dx, dx
b g (b)

i4 =
i=1

n(2n + 1)(n +

1)(3n2 30

+ 3n − 1) 1.

f (g (x))g (x) dx ⇒
a a

f (u) du ⇒
g (a)

f (u) du

a

f (x) dx = 0
a a b

Numerical Integration f (x) dx
a n

Upper and Lower Sums a du = 1 ln a a + C, a > 0
u n

2.
b b

f (x) dx = −
c

Midpoint Rule: f (x) dx ≈
b i=1

f

I.R. = lim

n→∞

f (mi )∆x
i=1

xi + xi−1 2
n

∆x

1 du = ln |u| + C u tan u du = − ln | cos x| + C

3.
a b

f (x) dx =
a

f (x) +
c b

f (x) dx Trapezoidal Rule:

f (mi ) is the min. value of f on the subinterval. 4.
n

b

f (x) dx ≈
a b i=1

f (xi ) + f (xi−1 ) 2

∆x

kf (x) dx = k
a a

f (x) dx Simpson’s Rule:
a

C.R. = lim = ln | sec u| + C cot u du = ln | sin u| + C = − ln | csc u| + C sec u du = ln | sec u + tan u| + C = − ln | sec u − tan u| + C csc u du = − ln | csc u + cot u| + C = ln | csc u − cot u| + C

n→∞

f (Mi )∆x
i=1

f (x) dx ≈

5.

[f (x)dx ± g (x) ± · · · ± k(x)] dx =
b b b

b−a [f (x0 ) + 4f (x1 )+ 3n

f (Mi ) is the max. value of f on the subinterval. Left Endpoints: a + (i − 1)∆x, for i = 1, . . . , n Right Endpoints: a + i∆x, for i = 1, . . . , n ∆x = (b − a)/n

2f (x2 ) + 4f (x3 ) + · · · + 4f (xn−1 ) + f (xn )] k(x) dx

f (x) ±
a b a

g (x) dx ± · · · ±
a

6. 0 ≤
a

f (x) dx for f non-negative
b b

7. If f (x) ≤ g (x) ⇒
a

f (x) dx ≤
a a

g (x) dx
a

8. If f is an even function, then
−a a

f (x) dx = 2
0

f (x) dx

9. If f is an odd function, then
−a

f (x) dx = 0