Esercizi sugli Integrali

1.

∫x√x dx =

2

5x2√x+ c;

2.

∫1

3 + 3x2=

1

3arctanx+ c

3.

∫5

x2 − 1dx = −5

2ln

∣∣∣∣1 + x

1− x

∣∣∣∣+ c

4.

∫(x− 6)5 dx =

(x− 6)6

6+ c

5.

∫e5x−1 dx =

1

5e5x−1 + c

6.

∫3

1 + 9x2dx = arctan 3x+ c

7.

∫sinx cosx dx =

1

2sin2 x+ c

8.

∫2x− 3

(x2 − 3x+ 1)2dx = − 1

x2 − 3x+ 1+ c

9.

∫cosx

sin3 xdx = − 1

2 sin2 x+ c

10.

∫ln3 x

xdx =

1

4ln4 x+ c

11.

∫x2

x3 − 9dx = ln |x3 − 9|+ c

12.

∫x− 1

1 + x2dx =

1

2ln(1 + x2)− arctanx+ c

13.

∫x2 3√x3 − 4 dx =

1

4(x3 − 4) · 3

√x3 − 4 + c

14.

∫3x+ 2

1− x2dx = −3

2ln |1− x2|+ ln

∣∣∣∣1 + x

1− x

∣∣∣∣+ c

15.

∫ex − e−x

ex + e−xdx = ln(ex + e−x) + c

16.

∫cosx

1 + sin2 xdx = arctan sinx+ c

17.

∫x3 ln(x− 2) dx =

1

4x4 ln(x− 2)− x4

16− x3

6− x2

2− 2x− 4 ln(x− 2)+ c

18.

∫sin lnx dx =

1

2x(sin lnx− cos lnx) + c

19.

∫x2e3x dx =

1

27e3x(9x2 − 6x+ 2) + c

20.

∫x3 lnx dx =

x4

16(4 lnx− 1) + c

21.

∫x · arctanx dx =

1

2(x2 arctanx− x+ arctanx) + c

22.

∫(x+1) ln(x+2) dx =

1

4[2(x+1)2 ln(x+2)−(x+2)2+4x−2 ln(x+2)]+c

23.

∫x3 + 8

x− 2dx =

x3

3+ x2 + 4x+ 16 ln |x− 2|+ c

24.

∫x3 ln(3x+1) dx =

x4 ln(3x+ 1)

4− x4

16+x3

36− x2

72+

x

108− ln(3x+ 1)

324+c

25.

∫x2 ln(x2 + 1) dx =

x3 ln(x2 + 1)

3− 2x3

9+

2x

3− 2

3arctanx+ c

26.

∫x3 arctanx dx =

x4 arctanx

4− x3

12+

x

4− arctanx

4+ c

27.

∫arctan

(x+ 1

1− x

)dx = x arctan

(x+ 1

1− x

)− ln(x2 + 1)

2+ c

28.

∫ln(1−

√x) dx = x ln(1−

√x)− ln(1−

√x)−

√x(√x+ 2)

4+ c

29.

∫ √x− 2 ln(x2−4x+3) dx =

2√

(x− 2)3

3ln(x2−4x+3)−4

3arctan

√x− 2−

2

3ln

∣∣∣∣√x− 2− 1√x− 2 + 1

∣∣∣∣− 8√(x− 2)3

9+ c

30.

∫ln

(1 +

1

x2

)dx = x ln

(1 +

1

x2

)+ 2arctanx+ c

31.

∫ √1 +√x

xdx = 4

√1 +√x+ 2 ln

∣∣∣∣∣√

1 +√x− 1√

1 +√x+ 1

∣∣∣∣∣+ c

32.

∫ln(x2 + 2) dx = x ln(x2 + 2)− 2x+ 2

√2 arctan

x√2

2+ c

33.

∫lnx

(x+ 1)2dx = − lnx

x+ 1+ ln

(x

x+ 1

)+ c