1. 11 1( ) ( )1

n nax b dx ax bn a

, 1n

2. ln | |dx xx

3. 1 ln | |dx ax bax b a

4. 1ax axae dx e , 1

lnx x

aa dx a

5. ln (ln 1)x dx x x

6. 1sin( ) cos( )aax b dx ax b

7. 1cos( ) sin( )aax b dx ax b

8. 2 cos sinsin2 2x x xx dx

9. 2 cos sincos2 2x x xx dx

10. 1 21 1sin sin cos sinn n nnx dx x x x dxn n

11. 1 21 1cos cos sin cosn n nnx dx x x x dxn n

12. tg ln | cos |x dx x

13. ctg ln | sin |dx x

14. 1 21tg tg tg1

n n nx dx x x dxn

15. 1 21ctg ctg ctg1

n n nx dx x x dxn

16. 1ln | tg |cos sin

dx xx x

17. 1ln | ctg |sin sindx x

x x

18. 2 2

1 1 tg 2 11 1cos cos cosn n n

x ndx dxn nx x x

19. 2 2

1 1 ctg 2 11 1sin sin sinn n n

x ndx dxn nx x x

20. 2arcsin arcsin 1x dx x x x

21. 2arccos arccos 1x dx x x x

22. 2arctg arctg ln 1x dx x x x

23. 2arcctg arcctg ln 1x dx x x x

24. 2 2 2 2 11 1( ) ( ) , 12 1

n nx x a dx x a nn

25. 2 22 2 lnxdx x a

x a

26. 2 2

1 arctgdx xa ax a

27. 122 2 ln | |a

dx x ax ax a

28. 2 2 2 2 313 ( )x x a dx x a

29. 2 2

2 2

xdx x ax a

30. 2 2 2 2 2 2 212 ln | |x a dx x x a a x x a

31. 2 2

2 2ln | |dx x x a

x a

32. 2 2

2 2

1 (ln ln( ))dx x x a aax x a

33. 2 2

2 2

xdx a xa x

34. 2 2 2 2 21 arcsin2

( )xa x dx x a x aa

35. 2 2

arcsindx xaa x

36. 21 ( 1)ax ax

axe dx e ax

37. 11n ax n ax n axna ax e dx x e x e dx

38. 2 21sin ( sin cos )ax ax

a be bx dx e a bx b bx

39. 2 21cos ( cos sin )ax ax

a be bx dx e a bx b bx

40. 1sinh( ) cosh( )aax dx ax

41. 1cosh( ) sinh( )aax dx ax

42. tanh( ) ln | cosh( ) |x dx x

Area with Double Integral: 2

12 1( ) ( )

x

A xA dxdy y x y x dx

(standard formula: y2 is top function, y1 is bottom function) Volume with Triple Integral: 2 1( , ) ( , )

xyV DV dxdydz z x y z x y dxdy (analogous)

Change of coordinates in integral

Polar 2D cossin

x ry r

[0, ], [0, 2 ]r a

dxdy rdrd

Cylindrical 3D cossin

x ry rz z

1 2[0, ], [0,2 ], [ , ]r a z z z dx dy dz r dr d dz

Spherical 3D cos sinsin sincos

xyz

[0, ], [0, 2 ], [0, ]r a 2 sindx dy dz d d d

Parameterize – Curves (x(t),y(t),z(t) and Surfaces (x(u,v),y(u,v),z(u,v))

Curves (1 parameter) Surfaces (2 parameters) Explicit function y=y(x)

( )x ty y t

[0, 2 ]t

Disk at z=b with radius R; center (p,q) ( , ) cos( )

[0, 2 ]( , ) sin( )

[0, ]( , )

x u v v u pu

y u v v u qv R

z u v b

Circle radius R; center (p,q) cos

sin

x t R t p

y t R t q

[0, 2 ]t

Cylinder with radius R and center (p,q)

1 2

( , ) cos( )[0, 2 ]

( , ) sin( )[ , ]

( , )

x t z R t pt

y t z R t qz z z

z t z z

Ellipse center (p,q); a and b cos

sin

x t a t p

y t b t q[0, 2 ]t

Sphere with radius R and center (p,q,s) ( , ) cos( )sin( )( , ) sin( )sin( )( , ) cos( )

x u v R u v py u v R u v qz u v R v s

[0,2 ][0, ]

uv

2 2( ) ( )az x p y q , z b (cut off paraboloid)

( , ) cos( )[0, 2 ]

( , ) sin( )[0, ]

( , )

x t z az t pt

y t z az t qz b

z t z z

Line Integrals: Let C: (x(t),y(t),z(t)) parameterized curve with endpoints 1 2[ , ]t t

I. Arc length ds: 2

1

2 2 2( , , ) ( ( ), ( ), ( ))t

tC

f x y z ds f x t y t z t x y z dt

2D: 2

1

2 2( , ) ( ( ), ( ))t

tC

f x y ds f x t y t x y dt If y=y(x), parameterize with x=t and y=y(t).

II. *With dx, dy, dz : 2

1

( , , ) ( , , ) ( , , ) ( )t

tC

P x y z dx Q x y z dy R x y z dz P x Q y R z dt

**SPECIAL If rot[<P,Q,R>]=0 then find potential u: ( , )u P u Pdx f y zx

…

( , , ) ( , , ) ( , , ) ( ) ( )C

P x y z dx Q x y z dy R x y z dz u B u A ; A and B are endpoints of C.

2D: 2

1

( , ) ( , ) ( )t

tC

P x y dx Q x y dy P x Q y dt (set R=0 in all 3D formulas!)

Green’s theorem for closed curve C on x0y plane enclosing surface D.

( , ) ( , )DC

Q PP x y dx Q x y dy dxdyx y

(+ means D is on your “left” as you walk C)

Surface Integrals: I. S= (x(u,v),y(u,v),z(u,v)) ( , , ) ( ( , ), ( , ), ( , ))

uv

u vSD

f x y z dS f x u v y u v z u v dS dS du dv

where , ,uyx zdS u u u and , ,v

yx zdS v v v

S: z=z(x,y) 2 2( , , ) ( , , ( , )) ( ) ( ) 1xy

SD

z zf x y z dS f x y z x y dx dyx y

II. With dydz, dxdz, dxdy: ( , , ) ( , , ) ( , , ) , , ( )

uvu vS D

P x y z dydz Q x y z dxdz R x y z dxdy P Q R dS dS du dv

S: z=z(x,y) , , , ,1xyS D

z zPdydz Qdxdz Rdxdy P Q R dxdyx y

S: F(x,y,z)=c , , , ,1xy

FF yx

F FS Dz z

Pdydz Qdxdz Rdxdy P Q R dx dy

*** Stokes’ theorem for closed curve C in space enclosing surface S:

SC

dydz dxdz dxdy

Pdx Qdy Rdz x y zP Q R

(+ means S is on your left as you walk C with your head pointed in +z direction)

If we have 2D, then R=0 and Green’s thm.

****Gauss- Ostrogradsky’s theorem for closed surface S enclosing solid G

( , , ) ( , , ) ( , , )S G

P Q RP x y z dydz Q x y z dxdz R x y z dxdy dx dy dzx y z

Vector Calculus Scalar function: y=f(x), z=f(x,y), w=f(x,y,z) are scalar (real-valued functions) Vector field has components that are scalar functions.

1D: ( )x xF r x r i , 2D: ( , ), ( , )x y x yF r x y r x y r i r j ,

3D: ( , , ), ( , , ), ( , , )x y z x y zF r x y z r x y z r x y z r i r j r k

Note: ( , , )z zr r x y z is the z-component of the vector and it is a scalar function of the 3 independent variables x,y,z (Analogous for xr and yr )

Gradient, divergence and rotation are operators.

Gradient: scalar function vector field : nabla or del

f=f(x,y,z) grad( ) , ,f f f f f ff f i j kx y z x y z

Divergence: vector field scalar function

, ,x y zF r r r ( ) yx zrr rdiv F

x y z

We write: ( ) , , , ,x y zdiv F F r r rx y z

“scalar” product

Rotation (curl): vector field vector field

, ,x y zF r r r ( )

x y z

i j k

rot F Fx y z

r r r

“vector” product

2D: ,x yF r r use 0zr

Work: Let F be a vector field; C a curve parameterized by t, ( ( ), ( ), ( ))C x t y t z t , 1 2[ , ]t t t

*Then 2

1( ) ( ), ( ), ( )

t

C tW F ds F t x t y t z t dt (Analogous 1D:

b

a

W Fds )

Potential: Let F be a vector field with ( ) 0rot F . Then there exists f such that f F .

** ( ) ( )C C

W F ds f ds f B f A where A and B are the endpoints of C.

f is called the potential of F .

Circulation (Stokes): Let F and C a closed curve bounding the surface S. *** ( )

SC

Circ F ds rot F ndS , where n is the unit normal pointing “out” from C.

Divergence (Gauss) Theorem: Let F and closed surface S enclosing solid G. **** ( )

S GFlux FdS div F dxdydz