M141 Summary sheet and Table of integral PDF

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TABLE OF INTEGRALS [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

[13]

[14]

[15] [16]

[17]

[18] [19] [20]

[21]

[22]

[23]

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

Z

xn dx =

1 xn+1 + c, n 6= −1 n+1

dx = ln |x| + c x ex dx = ex + c sin x dx = − cos x + c cos x dx = sin x + c tan x dx = ln | sec x| + c sec2 x dx = tan x + c cosec2 x dx = − cot x + c sinh x dx = cosh x + c cosh x dx = sinh x + c tanh x dx = ln(cosh x) + c

  b 1 n+1 ax + b − (ax + b) + c, n 6= −1, −2 n+2 a2 n+1   Z x2 1 1 dx = 3 (ax + b)2 − 2b(ax + b) + b2 ln |ax + b| + c ax + b a 2   Z x2 1 b2 dx = − 2b ln |ax + b| +c ax + b − (ax + b)2 ax + b a3   Z √ 2 (ax + b)5/2 b(ax + b)3/2 x ax + b dx = 2 +c − a 3 5 Z x 2ax − 4b√ √ ax + b + c dx = 3a2 ax + b   Z  √ax + b − √b  1 1   √  + c, b > 0 √ dx = √ ln  √ x ax + b b  ax + b + b  Z x dx √ = sin−1 +c 2 2 a a −x Z   dx 1 −1 x = +c tan a2 + x2 a a   Z 1 1  x + a  +c ln dx = 2a  x − a  a2 − x2   Z x + a 1 1 x   +c + 3 ln  dx = 2 2 4a x − a 2a (a − x2 ) (a2 − x2 )2   Z  a + √a2 − x2  1 1p 2   a − x2 dx = − ln  +c x  a  x Z x 1 1 dx = + Z

x(ax + b)n dx =

[24] [25]

[26]

[27] [28]

[29]

[30] [31] [32] [33] [34] [35]

[36]

[37] [38] [39] [40]

√ 2 2 a+ a −x    p  2 − x2 − a ln  dx = a +c  Z  p   x x  2 2 √ 1 + x ± a dx = ln +c x x2 ± a2   √ Z 1  a + x2 + a2  1 √ dx = − ln  +c x x2 + a2  a  x Z x 1 1 dx = ± 2 √ +c 2 a (x2 ± a2 )3/2 x ± a2 Z p   p 1 p 1   x2 ± a2 dx = x x2 ± a2 ± a2 ln x + x2 ± a2  + c 2 2   √ √ Z   p x2 + a2  a + x2 + a2  2 2 dx = x + a − a ln  +c   x x Z  1 1 ax − ln(b + keax ) + c, ab 6= 0 dx = ax ab b + ke Z 1 eax sin bx dx = 2 eax (a sin bx − b cos bx) + c a + b2 Z 1 eax cos bx dx = 2 eax (a cos bx + b sin bx) + c a + b2 Z Z 1 n−1 sinn x dx = − cos x sinn−1 x + sinn−2 x dx n n Z Z 1 n−1 cosn x dx = sin x cosn−1 x + cosn−2 x dx n n Z Z 1 tann x dx = tann−1 x − tann−2 x dx n−1 Z Z secn−2 x tan x n − 2 + secn−2 x dx secn x dx = n−1 n−1 Z Z n−1 sinm+1 x cosn−1 x m n + sinm x cosn−2 x dx sin x cos x dx = m+n m+n Z Z xn ex dx = xn ex − n xn−1 ex dx

Z √ 2 a − x2

Z

Z

xn sin x dx = −xn cos x + n n

n

x cos x dx = x sin x − n

Z

Z

xn−1 cos x dx

xn−1 sin x dx

MATH141 Formula Sheet Final Exam: Strand 1 Hyperbolic Functions Definition

dy

1 1 = ′ = f ′ (f −1 (x)) dx f (y) Parametric Differentiation If x = f (t) and y = g(t), then dx = f ′ (t) dt

ex − e−x sinh x = 2 ex + e−x cosh x = 2

and dy = g ′ (t) dt

and moreover, g ′ (t) dt g ′ (t) dy = ′ = ′ dx f (t) dt f (t)

Identity

n-th Derivative of y = f (x)

cosh2 x − sinh2 = 1

dn y dn y d (n−1) f (x) = f (n) (x) and = n dx dx dxn where n is a natural number.

Differentiation If y = f (x), then

Optimal values

dy = df (x) = f ′ (x) dx

and

dy = f ′ (x) dx

Rules for Basic Functions Differential

Derivative (mx + b)′ = m

d(mx + b) = m dx

(xr )′ = rxr−1 (ex )′ = ex 1 (ln x)′ = x (sin x)′ = cos x

d(xr ) = rxr−1 dx d(ex ) = ex dx 1 d(ln x) = dx x d(sin x) = cos x dx

(cos x)′ = − sin x

d(cos x) = − sin x dx

Rules for Function Operations Let f and g be functions and k be a constant. Derivative ′

in the other word,

Differential d(kf ) = k df d(f + g) = df + dg

′

(kf ) = kf (f + g)′ = f ′ + g ′ (f g )′ = f ′ g + f g ′  f ′ f′ f g′ = − 2 g g g

d(f g) = g df + f dg f  1 f = df − 2 dg d g g g df (g) = f ′ (g) dg

Chain Rule If y = f (z) and z = g(x), then dy = f ′ (z) dz = f ′ (z)g ′ (x) dx

• The function f (x) has a local maximum at x = a provided f ′ (a) = 0 and f ′′ (a) < 0; • The function f (x) has a local minimum at x = a provided f ′ (a) = 0 and f ′′ (a) > 0. Integration Definition • indefinite integral • definite integral Basic Rules

Z

Z

f (x) dx

b

f (x) dx

a

• scalar-multiple rule Z Z kf (x) dx = k f (x) dx where k is a constant.

• sum rule Z Z Z (f (x) + g(x)) dx = f (x) dx + g(x) dx • simple substitution Z f ′ (g (x))g ′ (x) dx = f (g(x)) + constant The Fundamental Theorem of Calculus Z b f (x) dx = F (b) − F (a) a

in the other word,

where dy = f ′ (g (x))g ′ (x) dx

Differentiation of Function Inverses If y = f −1 (x), then x = f (y) and dx = f ′ (y) dy

F (x) =

Z

f (x) dx

Differentiation of Parametric Integral Z u(x) d f (t) dt = f (u(x))u′ (x) − f (v(x))v ′ (x) dx v(x)

MATH141 Calculus Formula Sheet Algebraic identities

(a ± b)2 2

=

a3 ± b3

=

a −b n

n

a −b

a2 ± 2ab + b2

=

2

Logarithmic rules

loga (x1 x2 ) = log a x1 + loga x2   x1 loga = loga x1 − log a x2 x2 n loga (x ) = n log a x

(a − b)(a + b)

(a ± b)(a2 ∓ ab + b2 )

(a − b)(an−1 + an−2 b + · · · + abn−2 + bn−1 )

=

Roots of a quadratic equation: The general form of the equation is

=

√ −b ± b2 − 4ac x= 2a • if b2 > 4ac, two real roots • if b2 = 4ac, two identical real roots • if b2 < 4ac, no real roots Radian and degree conversion

π radians

|x| =

aloga x

ax loga y log b x = logb a

loga x Modulus

and its solution is given by

=

=

x

y

ax2 + bx + c = 0, a, b, c ∈ R, a 6= 0

180◦

x



x, −x,

x ≥ 0; x < 0.

Even function

f (−x) = f (x) Odd function

f (−x) = −f (x)

Complex numbers Definition

• Standard from of a complex number: z = x + iy where x, y are real numbers, and i2 = −1.

Trigonometric identities

sin( π2 − θ)

cos θ

=

cos2 θ + sin2 θ

=

1

cos2 θ − sin2 θ

=

cos(2θ )

sin(−θ) =

• Its complex conjugate: z = x − iy Modulus and argument

− sin θ

|z| = r =

cos(−θ) = cos θ sin(2θ) = 2 sin θ cos θ cos(2θ) = 2 cos2 θ − 1 = 1 − 2 sin2 θ sin(θ ± φ) = sin θ cos φ ± cos θ sin φ

cos(θ ± φ) =

cos θ cos φ ∓ sin θ sin φ

p

x2 + y 2

arg z = θ,

where tan θ = y/x, x 6= 0 Polar form

z = r e iθ = r(cos θ + i sin θ) Properties

• product of a complex number and its conjugate zz = |z|2 = x2 + y2 = r2

Trigonometric laws

c b a = = sin A sin B sin C

• complex conjugate of product of complex numbers (z1 z2 ) = z1 z2

2

=

2

=

a

b

c

2

=

2

2

b + c − 2bc cos A

a2 + c2 − 2ac cos B 2

2

a + b − 2ab cos C

Some Trigonometric Values

θ sin θ cos θ

π/6 1/2 √ 3/2

π/√4 1/ 2 √ 1/ 2

√π/3 3/2 1/2

De Moivre’s Theorem

z n = (r eiθ )n = rn einθ = rn [cos(nθ) + i sin(nθ )]. nth roots z

1 n

     θ 2kπ 2kπ 1 θ + + = r n cos + i sin n n n n

where n is a natural number, k = 0, 1, 2, 3.., n − 1.

MATH141 Vectors Formula Sheet

Let A(A1 , A2 , A3 ) and B(B1 , B2 , B3 ) be two points in R3 . Also, let a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) be two vectors in R3 . Then

• Length: |a| = • Unit vector:

q

a12 + a22 + a23 ;

b= a

a ; |a|

− − →

Vector between A and B is AB = (B1 − A1 , B2 − A2 , B3 − A3 )

Direction cosines (l, m, n) = ( b a1 , a b2 , a b3 )

• Scalar/Dot Product: a · b = |a| |b| cos θ • Vector/Cross Product:

• Scalar triple product:

   i a × b =  a1  b1

j a2 b2

where

    , 

k a3 b3

θ = ∡(a, b)

a⊥(a × b) and b⊥(a × b).

   a1 [ a, b, c ] = a · (b × c) =  b1  c1

• Component of a on b: ab = a ·bˆ.

Projection of a on b:

LINES: If (x0 , y0 , z0 ) is a point in line L and

a2 b2 c2

a3 b3 c3

     

− → b ab = ab c

a = (a1 , a2 , a3 ) is a vector parallel to L then

• Symmetric equations (Cartesian) form of a line x − x0 y − y0 z − z0 . = = a1 a2 a3

L: • Parametric equation form of a line L :

x = x0 + a1 t,

y = y0 + a2 t,

z = z0 + a3 t, where t ∈ R.

• Vector parametric equation form of a line  L:

     x x0 a1  y  =  y0  +  a2  t. z z0 a3

PLANES: If r0 = (x0 , y0 , z0 ) is a point in plane P with a = (a1 , a2 , a3 ) and b = (b1 , b2 , b3 ) are vectors parallel to plane P then

• Vector parametric form of a plane is given by P    x P :  y =  z where s, t ∈ R.

:

r = r0 + at + bs, i.e.      x0 a1 b1 y0  +  a2  t +  b2  s. z0 a3 b3

• Linear equation form of a plane, P : (r − r0 ) · n = 0, where n is the normal vector to plane P . This equation simplifies to ax + by + cz = d . • Let P1 and P2 be planes. P3 : P1 + λP2 = 0, for some value of λ, is a plane with common line of intersection with P1 and P2 .

MATH141 Algebra Formula Sheet Sigma Notation: For a < b and a, b ∈ Z then b X

xi = xa + . . . + xb .

i=a

Kronecker Delta:

δij

( 1 if i = j = 0 if i 6= j

Matrix Multiplication:

Am×n Bn×p = (AB )m×p . Inverse of a Matrix - Adjoint Method:

A−1 = where adj A is the adjoint of A. For a 2 × 2 matrix,

1 adj A |A|  A11 A21

adj A = =

A12 A22

T

where Aij is the cofactor associated with the element aij in A. This is given by:

Aij = (−1)i+j Mij where Mij denotes the minor associated with each element aij . The minor ( Mij ) is the resultant determinant which is found after deleting the i−th row and the j−th column of A. Rank: The Rank of a matrix is the number of non-zero rows when the matrix is reduced to row echelon form (REF). Solving Systems of Linear Equations: Consider a system of equations

Ax = b where the matrix A has an inverse matrix.

• Reduced Row Reduction Form (RREF):

The RREF method is commonly used to solve a system of equations and/or to determine the inverse of a matrix. Given a system of equations of the matrix form:

Ax = b where A has an inverse matrix then the RREF method reduces the combined matrix

to using valid row reduction operations.





A|I |b



I | A−1 | x



This method is commonly called Gaussian Elimination.

• Cramer’s Rule:

xi =

Dx D

where D = |A| and Dx is the determinant of A with the i−th column replaced by b....