матанализ 1 курс 2 семестр Малышкин / лекции / Неопределенные интегралы

4x2

(5x + 3)

−

5

1

−

1

2

3

1

1

dx = (20x3 +12x2 )x

2 dx =

20x2 +12x

2 dx = 20

x2 +12

x2 + C

2

3

2

x

x

4x

2

x2

x2 +1−1

1

dx = 4

dx =4

dx =4

1

−

dx =4x − 4arctg(x) + C

x

2

+

x

2

x

2

+1

x

2

+1

1

+1

2

2

2

3

1

1

5

3

1

5

3

1

x

+

3x + 4

(t −1)

+ 3(

t −1) + 4

t

+ t

+ 2

dt = t

+ t

+ 2t−

dt =

2

t

+

2

t

+ t

+ C =

2

(x+1)

2

(x +1)

dx t ==x+1

dt =

2

2

2

2

2

2

2

+

2

+ (x +1)2 + C

3

3

x +1

dt =dx

t

t

5

5

3x + 4

3x + 4

3(t −1) + 4

3t

4

3

1

d (t 2 + 2)

1

dx

=

dx

=t ==x+1

dt =

dt +

dt =

+ 2

dt =

x2 + 2x + 3

(x +1)2 + 2

t2 + 2

t 2 + 2

t 2 + 2

2

t 2 + 2

1

2

+1

dt =dx

2 t

3

2

1

t

3

2

t

3

2

x +1

=

ln(t

+ 2) + 2

2

d

=

ln(t

+ 2) + 2

2arctg

+ C =

ln(x

+ 2x + 3) + 2

2arctg

+ C

2

t

2

2

2

+1

2

2

2

2

4) Интегрирование по частям

u dv = uv − vdu

fgdx = Fg − Fg dx = Gf − Gf dx

Пример:

t2

t

t2

t

2

(x +1)2

(x +1)2

x ln(x +1)dx t ==x+1

(t −1) ln tdt =

−t ln t −

−1dt =

−t ln t −

−t + C =

− x −1 ln(x+1) −

− x + C

2

2

4

2

4

dt

=dx

2