Download Indefinite Integral and Substitution Rule

Indefinite Integral and Substitution Rule
MATH 115 (S1)1
1. TWO FACTS ABOUT ANTIDERIVATIVES.
(1) If F(x) is an antiderivative of f (x), then for any constant C, F(x) + C is an antiderivative of f (x) too.
For example, F(x) = x3 is an antiderivative of f (x) = 3x2 . F(x) + 5 = x3 + 5 is also an antiderivative of
f (x) = 3x2 . In fact, for any constant C, F(x) + C = x3 + C is an antiderivative of f (x) = 3x2 .
(2) If F1 (x) and F2 (x) are two antiderivatives of f (x), then there is an constant C, such that
F1 (x) = F2 (x) + C.
By above two facts, if F(x) is an antiderivative of f (x), then any antiderivative of f (x) can be represented
to be F(x)+C, where C is a constant.
2. AFTER-CLASS QUESTIONS.(I will give the answer on my website.)
∫
∫b
∫
(1) ( f (x)dx)′ = f (x)
(2) ( a f (x)dx)′ = 0
(3) f ′ (x)dx = f (x) + C
3. PROPERTIES OF INDEFINITE INTEGRALS. (Similar with the properties of Definite integral.)
∫
∫
∫
∫
(1) k f (x)dx = k f (x)dx, where k is a constant. Especially, [− f (x)]dx = − f (x)dx.
∫
∫
∫
(2) [ f (x) + g(x)]dx = f (x)dx + g(x)dx.
∫
∫
∫
(3) [ f (x) − g(x)]dx = f (x)dx − g(x)dx.
∫
∫
∫
Generally, [k1 f (x) + k2 g(x)]dx = k1 f (x)dx + k2 g(x)dx, where k1 and k2 are constants.
4. SOME
BASIC FORMULAS OF INDEFINITE INTEGRAL.
∫
(1) kdx = kx + C, where k is a constant.
∫
∫
∫
n+1
2
(2) xn dx = xn+1 + C, where n , −1. For example, xdx = x2 + C,
x2 dx =
∫
∫
(3) cos xdx = sin x + C,
sin xdx = − cos x + C.
∫
∫
(4) sec2 xdx = tan x + C,
csc2 xdx = − cot x + C.
∫
∫
(5) sec x tan xdx = sec x + C,
csc x cot xdx = − csc x + C.
x3
3
+ C,
∫
x3 dx =
x4
4
+ C.
The above formulas will be used FREQUENTLY, please keep in mind!
5. TWO REMARKS ABOUT INDEFINITE INTEGRAL.
(1) Indefinite Integral may
∫ have “different” results.
For example, evaluate sec2 x tan xdx. We have two methods to∫ evaluate this indefinite
integral.
∫
2
2
Method 1: Let u = tan x, then du = sec xdx, so we have sec x tan xdx = udu = 12 u2 + C =
1
tan2 x + C.
2
∫
∫
Method 2: Let u = sec x, then du = sec x tan xdx, so we have sec2 x tan xdx = udu = 12 u2 + C =
1
sec2 x + C.
2
The above two results look different. But in fact, they are the same, because by trigonometric identity
sec2 x = tan2 x+1. So the second result can be changed into 21 sec2 x+C = 12 (tan2 x+1)+C = 12 tan2 x+ 21 +C =
1
tan2 x + C. (Since C is an arbitrary constant, 21 + C run through all constant, so we can just replace 12 + C by
2
C.) So in fact, the results from two methods are the same.
1
c
⃝January
2010 by Long Yu
1
∫
(2) How to check your indefinite integral results?
f (x)dx = F(x) + C, just “Take derivative” with
′
F(x) + C. If (F(x)
+ C) = f (x), then you are right; otherwise, you are wrong. For example, if you are not
∫
sure whether (x2 − x) cos xdx = (x2 − x − 2) sin x + (2x − 1) cos x + C is right. To check it, just need take
derivative with the result
d
[(x2 − x − 2) sin x + (2x − 1) cos x + C]
dx
d
d
d
= dx [(x2 − x − 2) sin x] + dx
[(2x − 1) cos x] + dx
(C)
= [(2x − 1) sin x + (x2 − x − 2) cos x] + [2 cos x − (2x − 1) sin x] + 0
= (x2 − x) cos x
So it is right.
6. SUBSTITUTION RULE FOR INDEFINITE INTEGRAL.
∫
The steps of using substitution rule evaluate indefinite integral.
∫
f (g(x))g′ (x)dx = f (u)du
Let u = g(x), then, du = g′ (x)dx, “make substitution (change varialbe)”
= F(u) + C
Integrate f (u)
= F(g(x)) + C
Replace u by g(x).
7. SOME TIPS WHEN USING SUBSTITUTION RULE.
The most important point when using substitution rule is to figure out which function is g(x) in the
integrand. We don’t have shortcuts for this, what you need to do is Practice and Try. but we do have some
tips to help you.
(1) Try to find a composite function factor, determine∫ its “inside function”. Then let u =“inside
function”, this is the possible substitution. For example, (x2 + 1)50 xdx. So (x2 + 1)50 is a composite
function factor, x2 + 1 is the “inside function” of this composite function, so let u = x2 + 1.
(2) Try to find an antiderivative of one factor of the integrand (most of the time, this antiderivative
will
∫ be a simple function). Then let u = “this antiderivative”, this is the possible substitution. For example,
sec2 x tan3 xdx. It is not difficult to find sec2 x is the derivative of tan x, that means tan x is an antiderivative
of the factor sec2 x, (tan x is a simple function), so let u = tan x, this is the possible substitution.
8. SUBSTITUTION RULE FOR DEFINITE INTEGRAL.
The steps of using substitution rule evaluate definite integral.
∫ g(b)
∫b
f (g(x))g′ (x)dx = g(a) f (u)du
Let u = g(x), make substitution (change varialbe).
a
g(b)
= F(u)|g(a)
evaluate indefinite Integra of f (u).
= F(g(b)) − F(g(a))
Calculate the value of F(u) at g(b) and g(a).
Remarks:
x
a
b
u = g(x) g(a) g(b)
(2) g′ (x)dx = du. That means we should replace all g′ (x)dx by du. Most of the time dx = du is wrong
except that u = x + b (b is a constant).
(1) Change variable, change limits. Use substitution table:
9. SOME COMMON MISTAKES IN USING SUBSTITUTION RULE. The following 2 items are
wrong. Don’t make such mistakes!!!
∫b
∫ g(b)
(1) Just replace dx by du. a f (g(x))g′ (x)dx = g(a) f (u)g′ (u)du.
∫b
∫b
(2) After changed variable, didn’t change upper limit and lower limit. a f (g(x))g′ (x)dx = a f (u)du.
If you missed the paper notes, please download from
http://www.math.ualberta.ca/˜longyu/
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