CHE COURSE INTEGRATION 1

ENGR. JOHN LESTER MORILLO

(ENGINEERING MATHEMATICS)

BASIC RULES OF ALGEBRA1. Commutative Property

addition: a + b = b + amultiplication: a·b = b·a

2. Associative Propertyaddition: (a + b) + c = a + (b + c)multiplication: (a · b) · c = a · (b · c)

3. Identity Propertyaddition: a + 0= 0 + a = amultiplication: a · 1= 1 · a = a

BASIC RULES OF ALGEBRA

4. Inverse Propertyaddition: a + (-a) = (-a) + a = 0multiplication: a · (1/a) = (1/a) · a = 1

5. Distributive Property of Multiplication5. Distributive Property of Multiplicationa ( b + c ) = ab + ac

PROPERTIES OF EQUALITY

1. Reflexive Propertya = a

2. Symmetric Propertyif a = b, then b = aif a = b, then b = a

3. Transitive Propertyif a = b and b = c, then a = c

PROPERTIES OF EQUALITY

4. Zero Product Propertyif ab=0, then a = 0 or b = 0 or both

a and b = 05. Addition Property of Equality5. Addition Property of Equality

if a = b, then a + c = b + c6. Multiplication Property of Equality

if a = b, then ac = bc

PROPERTIES OF EXPONENT

1. aman = am+n

2. am /an = am-n

3. (am) n = amn

4. (ab) m = ambm4. (ab) = a b5. (a/b) m = am /bm

6. a-m = 1 /am

7. a0 = 1

PROPERTIES OF LOGARITHM

1. log (xy) = log x + log y2. log (x/y) = log x – log y3. log (x) = n log x4. log 1 = 04. log 1 = 05. log a x = log x / log a6. log e x = ln x

ARITHMETIC PROGRESSION

An = A1 + (n-1)dSn = (n/2)(A1 + An) Sn = (n/2)[2A1 + (n-1)d]

An : last termA1 : first termn : number of termsd : common difference

GEOMETRIC PROGRESSION

An = A1 + (n-1)d aman = am+n

Gn = G1(r)n-1

Sn = G1(rn -1) / (r-1)Infinite Geometric Series: Sn = G1 / (1-r)Infinite Geometric Series: Sn = G1 / (1-r)

Gn : last termG1 : first termn : number of termsr : common ratio

TRIGONOMETRIC FUNCTIONS

sin A = opposite side/ hypotenuse sidecos A = adjacent side/hypotenuse sidetan A = opposite side/ adjacent sidecsc A = hypotenuse side / opposite sidecsc A = hypotenuse side / opposite sidesec A = hypotenuse side /adjacent sidecot A = adjacent side/ opposite side

LAW OF SINEa/sinA = b/sinB = c/sinC

LAW OF COSINE (SSS, SAS)LAW OF COSINE (SSS, SAS)a2 = b2 + c2 – 2bccos(A)b2 = a2 + c2 – 2accos(B)c2 = a2 + b2 – 2abcos(C)

Pythagorean Relation

sin2 A + cos2 A = 1

csc2 A= 1 + cot2 A

sec2 A= 1 + tan2 A

Sum and Difference of Angles

sin (A +/- B) = sinAcosB +/- cosAsinBcos (A +/- B) = cosAcosB – sinAsinBtan (A +/- B) = (tanA +/- tanB)/(1 -/+ tanAtanB)Double angleDouble anglesin (2A) = 2sinAcosAcos (2A) = cos2 A - sin2 A = 2cos2 A – 1

= 1 - 2sin2 A tan (2A) = (2tanA)/(1 - tan2 A)

Area of Triangle

A = ½ abA = [s(s-a)(s-b)(s-c)] ½

A = ½ ab sinCA = ½ bc sinAA = ½ bc sinAA = ½ ac sinB

Triangle inscribed in a circleA = abc/4r

Triangle circumscribing a circleA = rs

Triangle with escribed circleA = r(s-a)

Circle

Area: A = π r2

Circumference: C = 2 πr = πdArc: s= rθArea of Sector: A = ½ r2 θArea of Sector: A = ½ r θArea of a segment: A =area of sector –area

of triangle= ½ r2 (θ – sin θ)

Quadrilateral

Square A = s2P = 4s

Rectangle A = LWP = 2(L + W)P = 2(L + W)

Rhombus A = bh = s2 sin θ = ½ d1d2

P = 4sTrapezoid A = (h/2) (b1 + b2)

Ellipse and Parabolic Segment

Ellipse A = πabParabolic Segment: A = 2/3 bh

REGULAR POLYGON

Sum of Interior Angle: S = (n-2)(180)Size of Interior Angle: IA = (n-2)(180)/nNumber of Diagonals: Diagonals = (n-3)(n/2)Area:Area:

A = ½ (Perimeter)(apothem)Perimeter:

P = n (side length)

REGULAR POLYGON

A = ¼ nL2 cot (180/n)

Regular Polygon circumscribing a circle:A = nr2 tan (180/n)P = 2nr tan (180/n)P = 2nr tan (180/n)

Regular Polygon inscribed in a circle:A = ½ nr2 sin (360/n)P = 2nr sin(180/n)

Volumes and Surface AreasCube: V = s3

SA = 6s2Rectangular Parallelepiped:

V = LWHSA = 2(LW + WH + LH)

Right Prism and Right CylinderV = BhSA = 2 B + Lateral Area

Lateral area = (base perimeter)(h)

Oblique Prism and CylinderV = Bh = KeSA = 2 B + Lateral AreaLateral Area = (perimeter of Lateral Area = (perimeter of

the right section) x lateral edge

Pyramid and Cone:V = 1/3 Bh

Truncated Prism:V = Bh AVE

Frustum:V = h/3 [B1 + B2 + (B1B2)½ ]

Prismatoid: (Prismoidal Formula)V = L/6 (A1 + 4 Am + A2)

Sphere: V = 4/3 π r3 = π d3 /6 SA = 4 π r2 =

Zone: A = 2πrh Spherical Segment: V = (π h2/3) (3r – h)Spherical Sector: V = 1/3 (Area of Zone) rTorus: V = 2 π2 Rr2 Torus: V = 2 π2 Rr2

A = 4π2Rr

Ellipsoid

V = 4/3 π abc

Prolate Spheroid:V = 4/3 π ab2V = 4/3 π ab

Oblate Spheroid:V = 4/3 π a2b

ANALYTICAL GEOMETRY

Circle – locus of pt that w/c moves so that it is equidistant from a fixed pt called center.

1. Gen. Equation x2+y2+Dx+Ey+F=02. Std Equation c(0,0) x2+y2=r2 2. Std Equation c(0,0) x +y =r3. Std Equation c(h,k) (x-h)2+(y-k)2=r2

Parabola – locus of a pt w/c moves so that it is always equidistant to a fixed pt called focus and to fixed straight line called directrix.

1. Gen. Equation1. Gen. EquationAxis parallel to the y axis: Ax2+Dx+Ey+F=0Axis parallel to the x-axis: Cy2+Dx+Ey+F=0

2. Std. Equation, v(0,0)Axis along the y-axis x2=4ayAxis along the x-axis y2=4ax

3. Std. Equation, v(h,k)3. Std. Equation, v(h,k)Axis parallel to y-axis (x-h) 2=4a(y-k)Axis parallel to x-axis (y-k) 2=4a(x-h)

4. Eccentricity : e=15. Length of Latus Rectum: LR=Ι 4a Ι6. Distance from vertex to directrix: a6. Distance from vertex to directrix: a7. Distance from vertex to focus: a8. Distance from focus to directrix: 2a

Ellipse – locus of a pt w/c moves so that the sum of the distances to the two fixed points called foci is constant and is equal to the length of the major axis (2a).

1. General Equation: Ax2+Cy2+Dx+Ey+F=01. General Equation: Ax2+Cy2+Dx+Ey+F=02. Std. Equation, c(0,0)Major axis along y-axis y2/a2 + x2/b2 = 1Major axis along x-axis x2/a2 + y2/b2 = 1

3. Std. equation, c(h,k)Major axis parallel to y-axis

(y-k)2/a2 + (x-h)2/b2 = 1Major axis parallel to x-axisMajor axis parallel to x-axis

(x-h)2/a2 + (y-k)2/b2 = 14. Relationship bet. a, b and c : a2=b2+c2

5. Eccentricity : e=c/a ( less than 1)6. Length of Latus Rectum : LR=2b2/a

7. Length of major axis : 2a8. Length of semi-major axis: a9. Length of minor axis : 2b10. Length of semi-minor axis:b10. Length of semi-minor axis:b11. Distance between foci: 2c12. Distance from the center to focus: c

Hyperbola – locus of a pt w/c moves so that the difference of its distance to the two fixed pts called foci is constant and is equal to the length of the transverse axis (2a).

1. General Equation: Ax2+Cy2+Dx+Ey+F=01. General Equation: Ax2+Cy2+Dx+Ey+F=02. Std. Equation, c(0,0)Transverse axis along the y-axis:y2/a2-x2/b2=1Transverse axis along the x-axis:x2/a2-y2/b2=1

3. Std. equation, c(h,k)Transverse axis parallel to y-axis

(y-k)2/a2 - (x-h)2/b2 = 1Transverse axis parallel to x-axisTransverse axis parallel to x-axis

(x-h)2/a2 - (y-k)2/b2 = 14. Relationship bet. a, b and c : c2=a2+b2

5. Eccentricity : e=c/a ( greater than 1)6. Length of Latus Rectum : LR=2b2/a

7. Length of transverse axis : 2a8. Length of semi-transverse axis: a9. Length of conjugate axis : 2b10. Length of semi-conjugate axis: b10. Length of semi-conjugate axis: b11. Distance between foci: 2c12. Distance from the center to focus: c

Polar Coordinates:1. x = rcosθ2. y = rsinθ3. r2 = x2 + y23. r = x + y4. θ=arctan (y/x)

Differential Calculus

Derivative of Algebraic Functions:1. d/dx(c) = 02. d/dx (x) = 13. d/dx (u±v) = du/dx ± dv/dx3. d/dx (u±v) = du/dx ± dv/dx4. d/dx (uv) = u dv/dx + v du/dx5. d/dx (u/v) = (v du/dx – u dv/dx) / v2

6. d/dx (u) n = n un-1 du/dx

Derivative of Exponential Functions:1. d/dx (a) u = au ln a du/dx2. d/dx (e) u = eu d/dx

Derivative of Logarithmic Function1. d/dx (log u) = 0.4343 (du/dx) / u2. d/dx (ln u) = (du/dx) / u3. d/dx(log b u) = log b (du/dx) / u

Derivative of trigonometric functions1. d/dx (sin u) = cos u du/dx2. d/dx (cos u) = -sin u du/dx3. d/dx (tan u) = sec2 u du/dx3. d/dx (tan u) = sec u du/dx4. d/dx (cot u) = -csc2 u du/dx5. d/dx (sec u) = sec u tan u du/dx6. d/dx (csc u) = -csc u cot u du/dx

Derivative of Inverse Trigonometric Functions:

1. d/dx (arcsin u) = 1/(1-u2) 1/2 (du/dx)2. d/dx (arccos u) = -1/(1-u2) 1/2 (du/dx)3. d/dx (arctan u) = 1/(1+u2) (du/dx)3. d/dx (arctan u) = 1/(1+u ) (du/dx)4. d/dx (arccot u) = -1/(1+u2) 1/2 (du/dx)5. d/dx (arcsec u) = 1/u(u2-1) 1/2 (du/dx)6. d/dx (arccsc u) = -1/u(u2-1) 1/2 (du/dx)

Critical Points:1. At maximum point

y’ = 0 and y” is negative2. At Minimun point2. At Minimun point

y’= 0 and y” is positive3. At point of Inflection

y” = 0

L’Hopital’s Rule:lim xàa f(x) / g(x) = f(a) / g(a) = 0/0 or ∞/∞

lim xàa f(x)/g(x) = lim xàa f’(x)/g’(x) = f’(a)/g’(a)

1. Differentiate separately the numerator and denominator.

2. Substitute the value of the limit to the variables

Integral CalculusBasic Integral:1. ∫ du = u + C2. ∫ a du = au + C3. ∫ un du = (un+1) / (n+1) + C3. ∫ u du = (u ) / (n+1) + C4. ∫ du / u = ln Ι u Ι + C

Exponential and Logarothmic Functions:1. ∫ e du = e + C2. ∫ au du = au / ln Ι a Ι + C3. ∫ ln u du = u ln Ι u Ι – u + C3. ∫ ln u du = u ln Ι u Ι – u + C

Trigonometric Functions:1. ∫ sin u du = -cos u + C2. ∫ cos u du = sin u + C3. ∫ tan u du = ln Ι sec u Ι + C 3. ∫ tan u du = ln Ι sec u Ι + C

= -ln Ι cos u Ι + C4. ∫ cot u du = ln Ι sin u Ι + C

= -ln Ι csc u Ι + C

5. ∫ sec u du = ln Ι sec u + tan u Ι + C6. ∫ csc u du = ln Ι csc u - cot u Ι + C7. ∫ sec2 u du = tan u + C8. ∫ csc2 u du = -cot u + C8. ∫ csc u du = -cot u + C9. ∫ sec u tan u du = sec u + C10. ∫ csc u cot u u du = -csc u + C

Inverse Trigonometric Functions1. ∫arcsin u du = u arcsin u + (1-u2) 1/2 + C2. ∫arccos u du = u arccos u - (1-u2) 1/2 + C3. ∫arctan u du = u arctan u - ln Ι(1-u2)Ι 1/2 +C3. ∫arctan u du = u arctan u - ln Ι(1-u )Ι +C4. ∫du/ (a2-u2) 1/2 = arcsin (u/a) + C5. ∫du/ (a2 +u 2)= (1/a) arctan (u/a) + C6. ∫du/ u(u2-u2) 1/2 = (1/a) arcsec (u/a) + C

Trigonometric Substitution:1. ∫ (a2-u2) 1/2 du

Let: u = a sinθ and 1 – sin2θ = cos2θ2. ∫ (a2+u2) 1/2 du2. ∫ (a +u ) du

Let: u = a tanθ and 1 + tan2θ = sec2θ3. ∫ (u2-a2) 1/2 du

Let: u = a secθ and sec2θ – 1 = tan2θ

Integration by Parts

1. ∫ u dv = uv - ∫v du

Wallis’ Formula:1. ∫ sinm θ cosn θ dθLower limit = 0 and upper limit = π/2

=Where α = π/2 if both m and n are even

= 1 if otherwise