Antiderivative and integral. Indefinite integral, its properties and calculation. Antiderivative and indefinite integral Abstract and presentation for the lesson indefinite integral
Anoshina O.V.
workshop for bachelors [stamp of the RF Ministry of Education] / V.S.
Shipachev; ed. A.N. Tikhonov. - 8th ed., Rev. and add. Moscow: Yurayt, 2015 .-- 447 p.
2. Shipachev V. S. Higher mathematics. Complete course: tutorial
for acad. bachelor degree [Grif UMO] / VS Shipachev; ed. A.
N. Tikhonova. - 4th ed., Rev. and add. - Moscow: Yurayt, 2015 .-- 608
from
3. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics
in exercises and tasks. [Text] / P.E. Danko, A.G. Popov, T. Ya.
Kozhevnikov. At 2 pm - M .: Higher school, 2007 .-- 304 + 415c.
Test. Performed in accordance with:
Tasks and guidelines for the implementation of control works
in the discipline "APPLIED MATHEMATICS", Yekaterinburg, FGAOU
VO "Russian State Professional and Pedagogical
University ", 2016 - 30s.
Select the test option by the last digit of the number
grade book.
2.
Exam
antiderivative function f x defined on
some interval if F x f x for
each x from this interval.
For example, the function cos x is
the antiderivative of the sin x function, since
cos x sin x. Obviously, if F x is the antiderivative
function f x, then F x C, where C is some constant, is also
antiderivative function f x.
If F x is any antiderivative
function f x, then any function of the form
Ф x F x C is also
antiderivative function f x and any
the antiderivative is representable in this form. Definition. The totality of all
antiderivatives of the function f x,
identified at some
interval is called
indefinite integral of
function f x on this interval and
denoted f x dx. If F x is some antiderivative of the function
f x, then they write f x dx F x C, although
it would be more correct to write f x dx F x C.
According to the established tradition, we will write
f x dx F x C.
Thus, the same symbol
f x dx will denote as all
the set of antiderivatives of the function f x,
and any element of this set.
the integrand, and its differential is the subintegral expression. Really:
1. (f (x) dx) (F (x) C) F (x) f (x);
2.d f (x) dx (f (x) dx) dx f (x) dx.
differential continuously (x)
of the differentiable function is equal to
this function up to a constant:
d (x) (x) dx (x) C,
since (x) is antiderivative for (x).
antiderivatives, then the function f1 x f 2 x
also has an antiderivative, and
f1 x f 2 x dx f1 x dx f 2 x dx;
5. Kf x dx K f x dx;
6.f x dx f x C;
7.f x x dx F x C.
a 1
x
2.x a dx
C, (a 1).
a 1
dx
3.ln x C.
x
x
a
4.a x dx
C.
ln a
5.e x dx e x C.
6.sin xdx cos x C.
7.cos xdx sin x C.
dx
8.2 ctgx C.
sin x
dx
9.2 tgx C.
cos x
dx
arctgx C.
10.
2
1 x
dx
arcsin x C.
1 x 2
dx
1
x
12.2 2 arctan C.
a
a
a x
13.
14.
15.
dx
a2 x2
x
arcsin C ..
a
dx
1
x a
ln
C
2
2
2a x a
x a
dx
1
a x
a 2 x 2 2a ln a x C.
dx
16.
x2 a
ln x x 2 a C.
17.shxdx chx C.
18.chxdx shx C.
19.
20.
dx
ch 2 x thx C.
dx
cthx C.
2
sh x
properties: 1
1.dx d (ax)
a
1
2.dx d (ax b),
a
1 2
3.xdx dx,
2
1 3
2
4.x dx dx.
3
Decision. In the table of integrals we find
cos xdx sin x C.
We transform this integral to a tabular one,
taking advantage of the fact that d ax adx.
Then:
d 5 x 1
\u003d cos 5 xd 5 x \u003d
cos 5xdx cos 5 x
5
5
1
\u003d sin 5 x C.
5
3x x 1 dx.
Decision. Since under the integral sign
the sum of four terms is found, then
expand the integral into the sum of four
integrals:
2
3
2
3
2
3
x
3
x
x
1
dx
x
dx
3
x
dx xdx dx.
x3
x4 x2
3
x C
3
4
2
use the following properties
integrals:
If f x dx F x C, then
f x b dx F x b C.
If f x dx F x C, then
1
f ax b dx F ax b C.
a
1
6
2
3
x
dx
2
3
x
C
.
3 6
5
The following integrals are taken by the method of integration by parts:
a) x n sin xdx, where n 1,2 ... k;
b) x n e x dx, where n 1,2 ... k;
c) x n arctgxdx, where n 0, 1, 2, ... k. ;
d) x n ln xdx, where n 0, 1, 2, ... k.
When calculating integrals a) and b), introduce
n 1
notation: x n u, then du nx dx, and, for example
sin xdx dv, then v cos x.
When calculating the integrals c), d) denote by u the function
arctgx, ln x, and for dv take x n dx.
Decision.
u x, du dx
=
x cos xdx
dv cos xdx, v sin x
x sin x sin xdx x sin x cos x C.
x ln xdx
dx
u ln x, du
x
x2
dv xdx, v
2
x2
x 2 dx
ln x
=
2
2 x
x2
1
x2
1 x2
ln x xdx
ln x
C.
=
2
2
2
2 2
directly pick the antiderivative
for f x we \u200b\u200bcannot, but we know that
she exists. You can often find
antiderivative, introducing a new variable,
according to the formula
f x dx f t t dt, where x t and t is a new
variable
ax b
dx,
x px q
containing a square trinomial in
denominator of the integrand
expressions. Such an integral is also taken
variable change method,
pre-highlighting in
the denominator is a full square.
2
dx
.
x 4x 5
Decision. Convert x 2 4 x 5,
2
selecting a complete square according to the formula a b 2 a 2 2ab b 2.
Then we get:
x2 4x 5 x2 2 x 2 4 4 5
x 2 2 2 x 4 1 x 2 2 1
x 2 t
dx
dx
dt
x t 2
2
2
2
x 2 1 dx dt
x 4x 5
t 1
arctgt C arctg x 2 C.
1 x
1 x
2
dx
tdt
1 t
2
x t, x t 2,
dx 2tdt
2
t2
1 t
2
dt
1 t
1 t
d (t 2 1)
t
2
1
2
2tdt
2
dt
ln (t 1) 2 dt 2
2
1 t
ln (t 2 1) 2t 2arctgt C
2
ln (x 1) 2 x 2arctg x C.
1 t 2 1
1 t
2
dt
the problem of finding the area of \u200b\u200ba curvilinear
trapezium.
Let on some interval be given
continuous function y f (x) 0
A task:
Plot its graph and find the F area of \u200b\u200bthe figure,
bounded by this curve, by two straight lines x \u003d a and x
\u003d b, and from below - a segment of the abscissa axis between the points
x \u003d a and x \u003d b. The figure aABb is called
curved trapezoid
f (x) dx
Under a definite integral
a
of a given continuous function f (x) on
this segment is understood
the corresponding increment of it
antiderivative, that is
F (b) F (a) F (x) /
b
a
The numbers a and b are the limits of integration,
- the interval of integration.
values \u200b\u200bof the antiderivative integrand
functions for upper and lower limits
integration.
Introducing the notation for the difference
b
F (b) F (a) F (x) / a
b
f (x) dx F (b) F (a)
a
Newton-Leibniz formula.
the designation of the variable of integration, i.e.
b
b
a
a
f (x) dx f (t) dt
where x and t are any letters.
2) A definite integral with the same
outside
integration is zero
a
f (x) dx F (a) F (a) 0
a 3) When interchanging the limits of integration
definite integral reverses sign
b
a
f (x) dx F (b) F (a) F (a) F (b) f (x) dx
a
b
(additivity property)
4) If the interval is divided into a finite number
partial intervals, then a definite integral,
taken over the interval is equal to the sum of certain
integrals taken over all its partial intervals.
b
c
b
f (x) dx f (x) dx
c
a
a
f (x) dx 5) Constant multiplier can be taken out
for the sign of a definite integral.
6) A definite integral of an algebraic
sums of a finite number of continuous
functions is equal to the same algebraic
the sum of definite integrals of these
functions.
integral.
b
f (x) dx f (t) (t) dt
a
a (), b (), (t)
Where
for t [; ], functions (t) and (t) are continuous on;
5
Example:
1
=
x 1dx
=
x 1 5
t 0 4
x 1 t
dt dx
4
0
3
2
t dt t 2
3
4
0
2
2
16
1
t t 40 4 2 0
5
3
3
3
3
Definition. Let the function f (x) be defined on
infinite interval, where b< + . Если
exists
b
lim
f (x) dx,
b
a
then this limit is called improper
integral of the function f (x) on the interval
}
Main literature
1. Shipachev V. S. Higher mathematics. Basic course: textbook andworkshop for bachelors [stamp of the RF Ministry of Education] / V.S.
Shipachev; ed. A.N. Tikhonov. - 8th ed., Rev. and add. Moscow: Yurayt, 2015 .-- 447 p.
2. Shipachev V. S. Higher mathematics. Complete course: tutorial
for acad. bachelor degree [Grif UMO] / VS Shipachev; ed. A.
N. Tikhonova. - 4th ed., Rev. and add. - Moscow: Yurayt, 2015 .-- 608
from
3. Danko P.E., Popov A.G., Kozhevnikova T.Ya. Higher mathematics
in exercises and tasks. [Text] / P.E. Danko, A.G. Popov, T. Ya.
Kozhevnikov. At 2 pm - M .: Higher school, 2007 .-- 304 + 415c.
Reporting
1.Test. Performed in accordance with:
Tasks and guidelines for the implementation of control works
in the discipline "APPLIED MATHEMATICS", Yekaterinburg, FGAOU
VO "Russian State Professional and Pedagogical
University ", 2016 - 30s.
Select the test option by the last digit of the number
grade book.
2.
Exam
Indefinite integral, its properties and calculation Antiderivative and indefinite integral
Definition. The function F x is calledantiderivative function f x defined on
some interval if F x f x for
each x from this interval.
For example, the function cos x is
the antiderivative of the sin x function, since
cos x sin x. Obviously, if F x is the antiderivative
function f x, then F x C, where C is some constant, is also
antiderivative function f x.
If F x is any antiderivative
function f x, then any function of the form
Ф x F x C is also
antiderivative function f x and any
the antiderivative is representable in this form. Definition. The totality of all
antiderivatives of the function f x,
identified at some
interval is called
indefinite integral of
function f x on this interval and
denoted f x dx. If F x is some antiderivative of the function
f x, then they write f x dx F x C, although
it would be more correct to write f x dx F x C.
According to the established tradition, we will write
f x dx F x C.
Thus, the same symbol
f x dx will denote as all
the set of antiderivatives of the function f x,
and any element of this set.
Integral properties
The derivative of the indefinite integral isthe integrand, and its differential is the subintegral expression. Really:
1. (f (x) dx) (F (x) C) F (x) f (x);
2.d f (x) dx (f (x) dx) dx f (x) dx.
Integral properties
3. Indefinite integral ofdifferential continuously (x)
of the differentiable function is equal to
this function up to a constant:
d (x) (x) dx (x) C,
since (x) is antiderivative for (x).
Integral properties
4. If the functions f1 x and f 2 x haveantiderivatives, then the function f1 x f 2 x
also has an antiderivative, and
f1 x f 2 x dx f1 x dx f 2 x dx;
5. Kf x dx K f x dx;
6.f x dx f x C;
7.f x x dx F x C.
1.dx x C.
a 1
x
2.x a dx
C, (a 1).
a 1
dx
3.ln x C.
x
x
a
4.a x dx
C.
ln a
5.e x dx e x C.
6.sin xdx cos x C.
7.cos xdx sin x C.
dx
8.2 ctgx C.
sin x
dx
9.2 tgx C.
cos x
dx
arctgx C.
10.
2
1 x
Indefinite Integral Table
11.dx
arcsin x C.
1 x 2
dx
1
x
12.2 2 arctan C.
a
a
a x
13.
14.
15.
dx
a2 x2
x
arcsin C ..
a
dx
1
x a
ln
C
2
2
2a x a
x a
dx
1
a x
a 2 x 2 2a ln a x C.
dx
16.
x2 a
ln x x 2 a C.
17.shxdx chx C.
18.chxdx shx C.
19.
20.
dx
ch 2 x thx C.
dx
cthx C.
2
sh x
Differential properties
When integrating, it is convenient to useproperties: 1
1.dx d (ax)
a
1
2.dx d (ax b),
a
1 2
3.xdx dx,
2
1 3
2
4.x dx dx.
3
Examples of
Example. Evaluate cos 5xdx.Decision. In the table of integrals we find
cos xdx sin x C.
We transform this integral to a tabular one,
taking advantage of the fact that d ax adx.
Then:
d 5 x 1
\u003d cos 5 xd 5 x \u003d
cos 5xdx cos 5 x
5
5
1
\u003d sin 5 x C.
5
Examples of
Example. Calculate x3x x 1 dx.
Decision. Since under the integral sign
the sum of four terms is found, then
expand the integral into the sum of four
integrals:
2
3
2
3
2
3
x
3
x
x
1
dx
x
dx
3
x
dx xdx dx.
x3
x4 x2
3
x C
3
4
2
Variable independence
When calculating integrals, it is convenientuse the following properties
integrals:
If f x dx F x C, then
f x b dx F x b C.
If f x dx F x C, then
1
f ax b dx F ax b C.
a
Example
Let's calculate1
6
2
3
x
dx
2
3
x
C
.
3 6
5
Integration methods Integration by parts
This method is based on the udv uv vdu formula.The following integrals are taken by the method of integration by parts:
a) x n sin xdx, where n 1,2 ... k;
b) x n e x dx, where n 1,2 ... k;
c) x n arctgxdx, where n 0, 1, 2, ... k. ;
d) x n ln xdx, where n 0, 1, 2, ... k.
When calculating integrals a) and b), introduce
n 1
notation: x n u, then du nx dx, and, for example
sin xdx dv, then v cos x.
When calculating the integrals c), d) denote by u the function
arctgx, ln x, and for dv take x n dx.
Examples of
Example. Evaluate x cos xdx.Decision.
u x, du dx
=
x cos xdx
dv cos xdx, v sin x
x sin x sin xdx x sin x cos x C.
Examples of
Example. Calculatex ln xdx
dx
u ln x, du
x
x2
dv xdx, v
2
x2
x 2 dx
ln x
=
2
2 x
x2
1
x2
1 x2
ln x xdx
ln x
C.
=
2
2
2
2 2
Variable replacement method
Let it be required to find f x dx, anddirectly pick the antiderivative
for f x we \u200b\u200bcannot, but we know that
she exists. You can often find
antiderivative, introducing a new variable,
according to the formula
f x dx f t t dt, where x t and t is a new
variable
Integration of functions containing a square trinomial
Consider the integralax b
dx,
x px q
containing a square trinomial in
denominator of the integrand
expressions. Such an integral is also taken
variable change method,
pre-highlighting in
the denominator is a full square.
2
Example
Calculatedx
.
x 4x 5
Decision. Convert x 2 4 x 5,
2
selecting a complete square according to the formula a b 2 a 2 2ab b 2.
Then we get:
x2 4x 5 x2 2 x 2 4 4 5
x 2 2 2 x 4 1 x 2 2 1
x 2 t
dx
dx
dt
x t 2
2
2
2
x 2 1 dx dt
x 4x 5
t 1
arctgt C arctg x 2 C.
Example
To find1 x
1 x
2
dx
tdt
1 t
2
x t, x t 2,
dx 2tdt
2
t2
1 t
2
dt
1 t
1 t
d (t 2 1)
t
2
1
2
2tdt
2
dt
ln (t 1) 2 dt 2
2
1 t
ln (t 2 1) 2t 2arctgt C
2
ln (x 1) 2 x 2arctg x C.
1 t 2 1
1 t
2
dt
A definite integral, its basic properties. Newton-Leibniz formula. Certain integral applications.
The concept of a definite integral is led bythe problem of finding the area of \u200b\u200ba curvilinear
trapezium.
Let on some interval be given
continuous function y f (x) 0
A task:
Plot its graph and find the F area of \u200b\u200bthe figure,
bounded by this curve, by two straight lines x \u003d a and x
\u003d b, and from below - a segment of the abscissa axis between the points
x \u003d a and x \u003d b. The figure aABb is called
curved trapezoid
Definition
bf (x) dx
Under a definite integral
a
of a given continuous function f (x) on
this segment is understood
the corresponding increment of it
antiderivative, that is
F (b) F (a) F (x) /
b
a
The numbers a and b are the limits of integration,
- the interval of integration.
Rule:
The definite integral is equal to the differencevalues \u200b\u200bof the antiderivative integrand
functions for upper and lower limits
integration.
Introducing the notation for the difference
b
F (b) F (a) F (x) / a
b
f (x) dx F (b) F (a)
a
Newton-Leibniz formula.
Basic properties of a definite integral.
1) The value of the definite integral does not depend onthe designation of the variable of integration, i.e.
b
b
a
a
f (x) dx f (t) dt
where x and t are any letters.
2) A definite integral with the same
outside
integration is zero
a
f (x) dx F (a) F (a) 0
a 3) When interchanging the limits of integration
definite integral reverses sign
b
a
f (x) dx F (b) F (a) F (a) F (b) f (x) dx
a
b
(additivity property)
4) If the interval is divided into a finite number
partial intervals, then a definite integral,
taken over the interval is equal to the sum of certain
integrals taken over all its partial intervals.
b
c
b
f (x) dx f (x) dx
c
a
a
f (x) dx 5) Constant multiplier can be taken out
for the sign of a definite integral.
6) A definite integral of an algebraic
sums of a finite number of continuous
functions is equal to the same algebraic
the sum of definite integrals of these
functions.
3. Change of variable in a definite integral.
3. Replacing a variable in a specificintegral.
b
f (x) dx f (t) (t) dt
a
a (), b (), (t)
Where
for t [; ], functions (t) and (t) are continuous on;
5
Example:
1
=
x 1dx
=
x 1 5
t 0 4
x 1 t
dt dx
4
0
3
2
t dt t 2
3
4
0
2
2
16
1
t t 40 4 2 0
5
3
3
3
3
Improper integrals.
Improper integrals.Definition. Let the function f (x) be defined on
infinite interval, where b< + . Если
exists
b
lim
f (x) dx,
b
a
then this limit is called improper
integral of the function f (x) on the interval
}
