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A: TABLE OF BASIC DERIVATIVES Let u = u(x) be a differentiable function of the independent variable x , that is u ′ (x) exists. (A) The Power Rule : d {u n } = nu n−1 . u ′ dx d { u } = 1 . u′ dx 2 u d {c} = 0 , c is a constant dx

Examples : d {(x 3 + 4x + 1) 3/4 } = 3 (x 3 + 4x + 1) −1/4 . (3x 2 + 4) 4 dx 1 d { 2 − 4x 2 + 7x 5 } = (−8x + 35x 4 ) 2 5 dx 2 2 − 4x + 7x d {π 6 } = 0 , since π ≅ 3. 14 is a constant. dx

(B) The Six Trigonometric Rules : d {sin(u)} = cos(u). u ′ dx d {cos(u)} = − sin(u). u ′ dx d {tan(u)} = sec 2 (u). u ′ dx d {cot(u)} = − csc 2 (u). u ′ dx d {sec(u)} = sec(u) tan(u). u ′ dx d {csc(u)} = − csc(u) cot(u). u ′ dx (C) The Six Hyperbolic Rules : d {sinh(u)} = cosh(u). u ′ dx d {cosh(u)} = sinh(u). u ′ dx d {tanh(u)} =sech 2 (u). u ′ dx d {coth(u)} = −csch 2 (u). u ′ dx d {sech(u)} = −sech(u) tanh(u). u ′ dx d {csch(u)} = −csch(u) coth(u). u ′ dx

Examples : d {sin(x 3 )} = cos(x 3 ). 3x 2 dx d {cos x )} = − sin( x ). 1 dx 2 x d {tan{ 5 )} = sec 2 (5x −2 ). (−10x −3 ) dx x2 d [cot{sin(2x)}] = − csc 2 {sin(2x)}. 2 cos(2x). dx d {sec( 4 x )} = sec( 4 x ) tan( 4 x ). 1 x −3/4 4 dx d {csc(8x − 7)} = − csc(8x − 7) cot(8x − 7). 8 dx Examples : d {sinh( 3 x )} = cosh( 3 x ). 1 x −2/3 3 dx d {cosh(sec(x)} = sinh{sec(x)}. sec(x) tan(x) dx d [tanh{x 3 + sin(x 2 )}] =sech 2 {x 3 + sin(x 2 )}. (3x 2 + 2x cos(x 2 )) dx d {coth( 1 + 2x)} = −csch 2 ( 1 + 2x). (− 1 + 2) x x dx x2 d {sech{9x)} = −sech(9x) tanh(9x). 9 dx d [csch{sinh(3x)}] = −csch{sinh(3x)} coth{sinh(3x)}. 3 cosh(3x) dx

(D) The Exponential & Logarithmic Rule : d {e u } = e u . u ′ dx d {ln| u |} = u ′ u dx d {a u } = a u . ln(a). u ′ , a ∈ R, a > 0, a ≠ 1 dx d {log | u |} = 1 u ′ , a dx ln(a) u

a ∈ R, a > 0, a ≠ 1

Examples : d {e } = e −x 3 . (−3x 2 ) dx d {ln| x 3 + 5x + 6 |} = 3x 2 + 5 dx x 3 + 5x + 6 d {2 sec(x) } = 2 sec(x) . ln(2). sec(x) tan(x) dx 2 d {log | tan(x) |} = 1 sec (x) 4 dx ln(4) tan(x) −x 3

(E) The Six Inverse Trigonometric Functions : d {sin −1 (u)} = u′ dx 1 − u2 d {cos −1 (u)} = − u′ dx 1 − u2 d {tan −1 (u)} = u ′ dx 1 + u2 d {cot −1 (u)} = − u ′ dx 1 + u2 d {sec −1 (u)} = u′ dx |u| u 2 − 1 d {csc −1 (u)} = − u′ dx |u| u 2 − 1 (F) The Inverse Hyperbolic Functions : d {sinh −1 (u)} = u′ dx 1 + u2 d {cosh −1 (u)} = dx

u′ u −1 2

d {tanh −1 (u)} = u ′ dx 1 − u2 (G) The Product and Quotient Rules : d {uv} = u ′ v + uv ′ dx d {ku} = ku ′ , k is a constant dx d { u } = u ′ v − uv ′ dx v v2

Examples : 8x d {sin −1 (4x 2 )} = dx 1 − 16x 4 d {cos −1 (3x)} = − 3 dx 1 − 9x 2 1 2 x d {tan −1 ( x )} = 1 = 1+x dx 2 x (1 + x) d {cot −1 (e x )} = − e x dx 1 + e 2x d [sec −1 (x 4 )} = 4x 3 4x 3 = dx |x 4 | x 8 − 1 x4 x8 − 1 d {csc −1 (2x)} = − 2 1 =− 2 dx |2x| 4x − 1 |x| 4x 2 − 1

d {sinh −1 (ln(x)} = dx

Examples : 1/x 1 + ln 2 (x)

5 25x 2 − 1 − 22 d {tanh −1 ( 2 )} = x = 2−2 x 4 dx x −4 1− 2 x

d {cosh −1 (5x)} = dx

Examples : d {x 3 ln(5x + 1)} = 3x 2 ln(5x + 1) + x 3 5 5x + 1 dx 3 2 d { x } = 1 d {x 3 } = 1 . 3x 2 = 3x 4 dx 4 4 dx 4 2 3 2 d { tan(2x) } = 2 sec (2x) .x − tan(2x) .3x dx x3 x6

In table above it is assumed that u = u(x) and v = v(x) are differentiable functions

B: TABLE OF BASIC INTEGRALS Let r , a , b , and β ∈ R , r ≠ −1 , a ≠ 0 , and β > 0. (A) The Power Rule : (ax + b) r+1 r (ax + b) dx = +C ∫ a(r + 1)

∫ dx = ∫ 1 dx = x + C ∫ ax1 + b dx = 2a

ax + b + C

Examples :

∫ x −5 dx = − 14 x −4 + C , ∫(3x − 1) −2 dx = ∫ 7dx = 7 ∫ dx = 7x + C ∫ x 1+ 4 dx = 2 x + 4 + C.

(B) The Six Trigonometric Rules : ∫ sin( ax + b))dx = − 1a cos( ax + b) + C ∫ cos( ax + b)dx = 1a sin( ax + b) + C ∫ tan( ax + b)dx = 1a ln|sec( ax + b)|+C ∫ cot( ax + b)dx = 1a ln|sin( ax + b)|+C ∫ sec( ax + b)dx = 1a ln|sec( ax + b) + tan( ax + b)|+C ∫ csc( ax + b)dx = 1a ln|csc( ax + b) − cot( ax + b)|+C

Examples : ∫ sin(9x − 2)dx = − 19 cos(9x − 2) + C ∫ cos(3x)dx = 13 sin(3x) + C ∫ tan(5w − 1)dw = 15 ln|sec(5w − 1)|+C ∫ cot(1 − 7u)du = − 17 ln|sin(1 − 7u)|+C ∫ sec(3x)dx = 13 ln|sec(3x) + tan(3x)|+C ∫ csc(2t)dt = 12 ln|csc(2t) − cot(2t)|+C

(C) Additional Trigonometric Rules :

∫ sec 2 ( ax + b)dx = 1a tan( ax + b) + C ∫ csc 2 ( ax + b)dx = − 1a cot( ax + b) + C ∫ sec( ax + b) tan( ax + b)dx = 1a sec( ax + b) + C ∫ csc( ax + b) cot( ax + b)dx = − 1a csc( ax + b) + C (D) The Six Hyperbolic Rules : ∫ sinh( ax + b)dx = 1a cosh( ax + b) + C ∫ cosh( ax + b)dx = 1a sinh( ax + b) + C ∫ tanh( ax + b)dx = 1a ln[cosh( ax + b)] + C ∫ coth( ax + b)dx = 1a ln|sinh( ax + b)|+C ∫ sech( ax + b)dx = 2a tan −1 (e ax+b ) + C ∫ csch( ax + b)dx = 1a ln|tanh( ax + b)/2|+C

(3x − 1) −1 +C −3

Examples

∫ sec 2 (2u/3)du = 32 tan(2u/3) + C 1 cot( w ) + C = −2 cot( w ) + C ∫ csc 2 ( w2 )dw = − 1/2 2 2 1 ∫ sec(3u) tan(3u)du = 3 sec(3u) + C ∫ csc(5x) cot(5x)dx = − 15 csc(5x) + C Examples

∫ sinh(2x − 7)dx = 12 cosh(2x − 7) + C ∫ cosh( 2x5 )dx = 52 sinh( 2x5 ) + C ∫ tanh(2u)du = 12 ln[cosh(2u)] + C ∫ coth(x + 3)dx = ln|sinh(x + 3)|+C ∫ sech(3x − 6)dx = 23 tan −1 (e 3x−6 ) ++C ∫ csch(10t)dt = 101 ln|tanh(5t)|+C

(E) Additional Hyperbolic Rules :

Examples

∫ sech 2 ( ax + b)dx = 1a tanh( ax + b) + C ∫ csch 2 ( ax + b)dx = − 1a coth( ax + b) + C

∫ sech 2 (4w)dw = 14 tanh(4w) + C ∫ csch 2 (2u)du = − 12 coth(2u) + C sech(3x)

∫ sech( ax + b) tanh( ax + b)dx = − 1a sech( ax + b) + C ∫ sech(3x) tanh(3x)dx = − 3 + C ∫ csch( ax + b) coth( ax + b)dx = − 1a csch( ax + b) + C ∫ csch( 3x ) coth( 3x )dx = −3csch(x/3) + C (F) Exponential /Logarithmic Rules : ∫ e ax+b dx = 1a e ax+b + C 1 . k ax+b + C , 0 < k ∈ R , k ≠ 1. ∫ k αx+β dx = a ln(k) ∫ ax1+ b dx = 1a ln| ax + b|+C

Examples :

∫ e 7x dx = 17 e 7x + C ∫ 2 10x−17 dx = 101ln 2 ∫

2 10x−17 + C

1 dx = 1 ln|2x − 3|+C 2 2x − 3

. (G) The Three Inverse Trigonometric Functions : ∫ β 21− x 2 dx = sin −1 ( βx ) + C

∫ ∫

1 dx = 1 tan −1 ( x ) + C β β β2 + x2 1 dx = 1 sec −1 ( x ) + C , β β x x2 − β2

Examples :

∫ ∫

x>β

(H) The Three Inverse Hyperbolic Functions : ∫ β 21+ x 2 dx = sinh −1 ( βx ) + C 2

∫

1 dx = 1 tan −1 ( x ) + C 3 + x2 3 3 1 dx = 1 sec −1 ( x ) + C , x > 2. 2 2 x x2 − 4 Examples :

1 dx = cosh −1 ( x ) + C β 2 x −β

∫

1 dx = sin −1 (x/4) + C 2 16 − x

∫

1 dx = 1 tanh −1 ( x ) + C , |x|< β 2 β β β −x

∫

1 dx = sinh −1 (x) + C 2 1+x

∫

1 dx = cosh −1 (x/ 5 ) + C x −5

∫

2

2

1 dx = 1 tanh −1 ( x ) + C 6 6 36 − x 2

(I) The Fundamental Theorems b

,

|x|< 6

Examples : e3

x=b x=e = g(b) − g(a) = ln(e 3 ) − ln(e) = 3 − 1 = 2 ∫ a f(x)dx = g(x)| x=a ∫ e 1x dx = ln|x| | x=e d { v(x) F(t) dt = F(v(x)). v ′ (x) − F(u(x)). u ′ (x) d { x cos(t 2 )dt = cos(x 4 ). 2x − cos(x 2 ). 1 ∫ u(x) dx dx ∫ x 3

2

In table above it is assumed that : (1) The function f(x) is continuous on [a, b] and ∫ f(x) dx = g(x) + C. (2) The functions u(x) and v(x) are differentiable and ∫

v(x) u(x)

F(t) dt exists.

C: BASIC TRIGONOMETRIC IDENTITIES GROUP (A) : (i) tan(θ) = (iii) sec(θ) =

sin(θ) cos(θ) 1 cos(θ)

(ii) cot(θ) =

cos(θ) sin(θ)

(iv) csc(θ) =

1 sin(θ)

GROUP (B) : (i) cos 2 (θ) + sin 2 (θ) = 1

(ii) 1 + tan 2 (θ) = sec 2 (θ)

(iii) cot 2 (θ) +1 = csc 2 (θ)

(ii) cos(2θ) = 2 cos 2 (θ) − 1

(iii) cos(2θ) = 1 − 2 sin 2 (θ)

GROUP (C) : (i) sin(2θ) = 2 sin(θ) cos(θ)

(iv) cos 2 (θ) = 1 [1 + cos(2θ)] 2

(v) sin 2 (θ) = 1 [1 − cos(2θ)] 2

GROUP (D) (i) sin(−θ) = − sin(θ)

(ii) cos(−θ) = cos(θ)

GROUP(E) (i) cos(θ ± φ) = cos(θ) cos(φ) ∓ sin(θ) sin(φ) (ii) sin(θ ± φ) = sin(θ) cos(φ) ± cos(θ) sin(φ) GROUP (F) (i) cos(θ) cos(φ) = 1 [cos(θ − φ) + cos(θ + φ)] 2 (ii) sin(θ) sin(φ) = 1 [cos(θ − φ) − cos(θ + φ)] 2 (iii) sin(θ) cos(φ) = 1 [sin(θ − φ) + sin(θ + φ)] 2

(iii) tan(−θ) = − tan(θ).

D: SPECIAL TRIGONOMETRIC EQUATIONS (i) sin(x) = 0  x = nπ

(ii) cos(x) = 0

x =

(2n − 1) π 2

where n is an integer : n = 0, ±1, ±2, ±3, . . .

E: HYPERBOLIC FUNCTIONS (i) sinh(x) = 1 [e x − e −x ] 2 (iv) coth(x) =

cosh(x) sinh(x)

(ii) cosh(x) = 1 [e x + e −x ] 2 (v) sech(x) =

(vii) cosh 2 (x) − sinh 2 (x) = 1

1 cosh(x)

(viii) 1 − tanh 2 (x) = sech 2 (x)

(iii) tanh(x) = (vi) csch(x) =

sinh(x) cosh(x) 1 sinh(x)

(ix) coth 2 (x) − 1 = csch 2 (x)

F:PROPERTIES OF LOGARITHMS Let

x

and

y

be positive real numbers. (ii) ln(x) − ln(y) = ln( xy )

(i) ln(x) + ln(y) = ln(xy) (iv) ln(e k ) = k

(v) e ln(x) = x

(iii) ln(x m ) = m ln(x). (vi) ln(1) = 0 , ln(e) = 1.

G:SPECIAL VALUES (i) sin(0) = 0

(ii) cos(0) = 1

(iii) tan(0) = 0

(iv) sinh(0) = 0

(v) cosh(0) = 1

(vi) tanh(0) = 0

(vii) sin(nπ) = 0 and cos(nπ) = (−1) n , provided that " n " is an integer.

END