KARAVALI INSTITUTE OF TECHNOLOGY, MANGALORE

BASIC MATHEMATICS FORMULA SHEET FOR I SEMESTER

Laws of Indices:

1. am · an = am+n

2.am

an= am−n

3. (am)n = amn

4. a0 = 1

5. n√a = a1/n

Laws of Logarithm:

1. log(mn) = logm+ log n

2. log(mn

)= logm− log n

3. log(mn) = n logm

4. log e = 1

5. log 1 = 0

Combinations:

• nCr =n!

(n− r)! r!

• nCr =n Cn−r Where n ≥ r

• n! = n · (n− 1) · (n− 2) · · · 3 · 2 · 1

• nC0 = 1; nC1 = n; nC2 =n(n− 1)

2!; nC3 =

n(n− 1)(n− 2)

3!; etc.

Partial fractions:Proper fraction: If degree of numerator is < degree of denominator.

1.f(x)

(x− α)(x− β)(x− γ)=

A

(x− α)+

B

(x− β)+

C

(x− γ)

2.f(x)

(x− α)2(x− β)=

A

(x− α)+

B

(x− α)2+

C

(x− β)

3.f(x)

(x− α)(px2 + qx+ r)=

A

(x− α)+

Bx+ C

px2 + qx+ r

Improper fraction: If degree of numerator is ≥ degree of denominator.

• f(x)

g(x)= Quotient +

Remainder

Divisor= Q+

F (x)

g(x).

HereF (x)

g(x)will be a proper fraction.

Trigonometry:

• sin2 θ + cos2 θ = 1

• 1 + tan2 θ = sec2 θ

• 1 + cot2 θ = csc2 θ

Trigonometric functions with standard angles:

θ 0◦ 30◦(π/6) 45◦(π/4) 60◦(π/3) 90◦(π/2) 180◦(π) 270◦(3π/2)

sin θ 0 1/2 1/√

2√

3/2 1 0 -1

cos θ 1√

3/2 1/√

2 1/2 0 -1 0

tan θ 0 1/√

3 1√

3 ∞ 0 ∞

sin(−θ) = − sin θ; cos(−θ) = cos θ; tan(−θ) = tan θ

sin(2nπ + θ) = sin θ; cos(2nπ + θ) = cos θ; tan(2nπ + θ) = tan θsin(π/2− θ) = cos θ; cos(π/2 + θ) = − sin θ; sin(π − θ) = sin θ; tan(π + θ) = tan θ

Compound angle formulae:

• sin(A+B) = sinA cosB + cosA sinB

• sin(A−B) = sinA cosB − cosA sinB

• cos(A+B) = cosA cosB − sinA sinB

• cos(A−B) = cosA cosB + sinA sinB

• tan(A+B) =tanA+ tanB

1− tanA tanB

• tan(A−B) =tanA− tanB

1 + tanA tanB

Convert product into sum or difference:

• sinA cosB =1

2[sin(A+B) + sin(A−B)]

• cosA sinB =1

2[sin(A+B)− sin(A−B)]

• cosA cosB =1

2[cos(A+B) + cos(A−B)]

• sinA sinB = −1

2[cos(A+B)− cos(A−B)]

Multiple angle formulae:

• sin 2A = 2 sinA cosA =2 tanA

1 + tan2A

• cos 2A = cos2A− sin2A = 1− 2 sin2A = 2 cos2A− 1 =1− tan2A

1 + tan2A

• sin 3A = 3 sinA− 4 sin3A

• cos 3A = 4 cos3A− 3 cosA

• tan 3A =3 tanA− tan3A

1− 3 tan2A

Convert sum or difference into product:

• sinC + sinD = 2 sin

(C +D

2

)· cos

(C −D

2

)• sinC − sinD = 2 cos

(C +D

2

)· sin

(C −D

2

)• cosC + cosD = 2 cos

(C +D

2

)· cos

(C −D

2

)• cosC − cosD = −2 sin

(C +D

2

)· sin

(C −D

2

)Hyperbolic functions:

• sinhx =ex − e−x

2

• coshx =ex + e−x

2

Hyperbolic identity:

• cosh2 x− sinh2 x = 1

Complex Trigonometry:

• sinx =eix − e−ix

2i

• cosx =eix + e−ix

2

Standard Limits:

• limx→a

(xn − an

x− a

)= nan−1, where n is any rational number.

• limx→0

sinx

x= 1

• limx→0

tanx

x= 1

• limx→0

(1 + x)1/x = e OR limx→∞

(1 +

1

x

)x= e

• limx→∞

(1 +

x

n

)n= ex

Differentiation:

limδx→0

δy

δx= lim

δx→0

f(x+ δx)− f(x)

δx=dy

dx= f ′(x) = y1(x) = y′(x) = D(y)

Rules of differentiation:Rule-1: Function of a function rule OR Chain rule:

If y = f [g(x)] thendy

dx= f ′[g(x)] · g′(x)

Further if y = f [g{h(x)}] thendy

dx= f ′[g{h(x)}] · g′{h(x)} · h′(x)

Rule-2: Product rule:d

dx(uv) = (uv)′ = u · v′ + v · u′

Rule-3: Quotient rule:d

dx

(uv

)=(uv

)′=v · u′ − u · v′

v2

List of derivatives of standard functions:

y = f(x)dy

dx= f ′(x) y = f(x)

dy

dx= f ′(x)

1. xn nxn−1 16. constant (c) 0

2. log x1

x17. aF (x)± bG(x) aF ′(x)± bG′(x)

3. ex ex 18. ax ax log x

4. sinx cosx 19. sinhx coshx

5. cosx − sinx 20. coshx sinhx

6. tanx sec2 x 21. tanhx sech2x

7. cotx − csc2 x 22. cothx − csch2x

8. secx secx tanx 23. sechx − sechx tanhx

9. cscx − cscx cotx 24. cschx − cschx cothx

10. sin−1 x1√

1− x225. sinh−1 x

1√x2 + 1

11. cos−1 x−1√

1− x226. cosh−1 x

1√x2 − 1

12. tan−1 x1

1 + x227. tanh−1 x

1

1− x2

13. cot−1 x−1

1 + x228. coth−1 x

−1

x2 − 1

14. sec−1 x1

x√x2 − 1

29. sech−1x−1

x√

1− x2

15. csc−1 x−1

x√x2 − 1

30. csch−1x−1

x√

1 + x2

List of Integrals:

f(x)∫f(x) dx f(x)

∫f(x) dx

(Functions) (Integrations) (Functions) (Integrations)

1. xnxn+1

n+ 1, (n 6= −1) 13.

1

x√x2 − 1

sec−1 x

(n is a constant)

2. axax

log a14. kf ′(x) kf(x)

3.1

xlog x 15. sinhx coshx

4. ex ex 16. coshx sinhx

5. sinx − cosx 17. sech2x tanhx

6. cosx sinx 18. csch2x − cothx

7. tanx log(secx) 19. sechx tanhx − sechx

8. cotx log(sinx) 20. cschx cothx − cschx

9. secx log(secx+ tanx) 21. sec2 x tanx

10. cscx log(cscx− cotx) 22. csc2 x − cotx= log

(tan

(x2

))11.

1√1− x2

sin−1 x 23. secx tanx secx

12.1

1 + x2tan−1 x 24. cscx cotx − cscx

Standard Integrals:

•∫eax sin(bx+ c) dx =

eax

a2 + b2[a sin(bx+ c)− b cos(bx+ c)]

•∫eax cos(bx+ c) dx =

eax

a2 + b2[a cos(bx+ c) + b sin(bx+ c)]

-PADMANABHA KAMATH