Derivatives and Integrals - Trig derivatives and integrals, inverse trig, exponential/log, and related formulas

Oct 13, 2026

Upload by noah

Midterm formula sheet compiling key results for calculus topics: trigonometric integrals and derivatives, inverse trigonometric derivatives, and standard exponential/log differentiation and integration rules. It also includes hyperbolic-function integral patterns, distance and projection formulas in coordinate geometry, and space-curve tools such as derivative rules, arclength, curvature, TNB (tangent-normal-binormal) frame definitions, and acceleration decomposition. Finally, it provides fundamental trigonometric identities involving sums, differences, double angles, and related transformations.

LanguageEnglish

CategoryPaper Templates

FormatPDF

Pages2

Reading Time1 min

AccessFree to download

Derivatives and Integrals

Trig integrals and trig derivatives

Inverse trig derivatives

Exponential/log and hyperbolic-function integrals

Distance and projections

Derivatives/Integrals of space curves, arclength, curvature, TNB frame, acceleration

Trig identities

Derivatives and Integrals Trig integrals: 1. R sinx dx = 􀀀 cos x + C 2. R cosx dx = sin x + C 3. R tanx dx = 􀀀 ln j cos xj + C 4. R sec x dx = ln j sec x + tan xj + C 5. R cotx dx = ln j sin xj + C 6. R csc x dx = ln j csc x 􀀀 cot xj + C 7. R sec2 x dx = tan x + C 8. R csc2 x dx = 􀀀 cotx + C 9. R sec x tan x dx = sec x + C 10. R cscx cotx dx = 􀀀 cscx + C Trig derivatives: 1. ~~d~~dx (sin(x)) = cos(x) 2. ~~d~~dx (cos(x)) = 􀀀 sin(x) 3. ~~d~~dx (tan(x)) = sec2 (x) 4. ~~d~~dx (csc(x)) = 􀀀 csc(x) cot(x) 5. ~~d~~dx (sec(x)) = sec(x) tan(x) 6. ~~d~~dx (cot(x)) = 􀀀 csc2 (x) Inverse trig derivatives: 1. ~~d~~dx 􀀀sin􀀀1 (x)􀀁 = p1x~~2~~ 2. ~~d~~dx (cos􀀀1(x)) = 􀀀 p1x~~2~~ 3. ~~d~~dx (tan􀀀1(x)) = 1x2 4. ~~d~~dx (csc􀀀1(x)) = 􀀀 xp~~1~~x~~2 ~~􀀀1 5. ~~d~~dx (sec􀀀1(x)) = xp~~1~~x~~2 ~~􀀀1 6. ~~d~~dx (cot􀀀1(x)) = 􀀀 1x2 Exponential/log: 1. ~~d~~dx (bx ) = bx ln b 2. R bx dx = l~~b~~nxb + C 3. R ekx dx = ~~e~~kkx + C Integrals involving p 1 􀀆 x2 1. R p1x~~2~~ dx = sinh􀀀1(x) + C 2. R px~~12~~􀀀1 dx = cosh􀀀1(x) + C 3. R 1~~ ~~􀀀~~1~~x2 dx = tanh􀀀1(x) + C 4. R j~~ ~~xjp~~1~~x~~2 ~~+1 dx = 􀀀 csch􀀀1(x) + C 5. R xp1~~1~~􀀀x~~2~~ dx = 􀀀 sech􀀀1(x) + C 6. R 1~~ ~~􀀀~~1~~x2 dx = coth􀀀1(x) + C 7. R 1~~ ~~􀀀~~1~~x2 dx = ~~1~~2 ln 􀀌 ~~x~~x~~1~~1 􀀌 + C 8. R x2~~ ~~1 dx = ~~1~~2 ln 􀀌 ~~x~~x~~1~~1 􀀌 + C 9. R px~~12~~+1 dx = ln(x +px2 + 1) + C 10. R px~~2~~ dx = ~~x~~2p 1 + x2 􀀀 ~~1~~2 ln(x +p 1 + x2 ) + C 11. R xp1~~1~~+x~~2~~ dx = 􀀀 ln 􀀌 p~~x~~2~~ +~~x~~1+1~~ 􀀌 + C 12. R x2~~ ~~px~~2~~ dx = 􀀀 p~~ 1+~~x~~x~~2 + C Distance Formula d((x1 ; y 1 ; z1 ) ; (x2 ; y2 ; z2 )) =p(x1 􀀀 x2 )2 + (y1 􀀀 y2 )2 + (z1 􀀀 z2 )2 Projections proja b = ~~a~~a􀀁􀀁~~b~~aa compa b = jproja bj Derivatives/Integrals of Space Curves ~~d~~dt hf (t); g (t); h (t)i = h ~~d~~dt f (t) ; ~~d~~dtg (t) ; ~~d~~dt h (t)i Rhf(t); g (t); h (t)i dt = hR f (t) dt; R g(t) dt; R h (t) dti Arclength s (t) =Rta jr0 (u)j du Curvature 􀀔 (t) = 􀀌 ~~d~~dTs 􀀌 = jj~~ ~~d~~d~~ddT~~t~~rt~~ ~~jj = j~~ ~~r0~~ ~~(jtr􀀂(tr)0j~~ ~~0~~3~~(t)j TNB frame T = unit tangent vector, N = j~~ ~~T0T0((tt))~~ ~~j~~ ~~ , B = T 􀀂 N = j~~ ~~r0r0((tt))~~ ~~􀀂􀀂r00r00((tt))~~ ~~j Acceleration a (t) = v0 (t)T (t) + 􀀔(t)v (t)2 N (t), where v = j v(t)j Trig Identities 1. sin2 􀀒 + cos2 􀀒 = 1 2. sin(x 􀀆 y) = sin x cos y 􀀆 cos x sin y 3. cos(x 􀀆 y) = cos x cos y 􀀇 sin x sin y 4. sin(2􀀒) = 2 sin 􀀒 cos 􀀒 5. cos(2􀀒) = cos2 􀀒 􀀀 sin2 􀀒 6. tan(2􀀒) = 1~~tan~~tan~~􀀒2~~􀀒 7. sin(􀀒=2) = 􀀆 q ~~ 1 ~~􀀀~~ c~~2~~os 􀀒~~ 8. cos(􀀒=2) = 􀀆 q ~~ 1+c~~2~~os 􀀒~~

FAQs

What are the standard trigonometric integral results included?

The sheet lists multiple integrals for sin x, cos x, tan x, sec x, cot x, and csc x, plus sec^2 x, csc^2 x, and mixed forms like sec x tan x and csc x cot x.

How are inverse trigonometric derivatives presented?

It provides derivative formulas for arcsin, arccos, arctan, and corresponding csc−1, sec−1, and cot−1 expressions, each written as a simplified algebraic form in x.

Which geometry and space-curve formulas are included?

It includes a 3D distance formula, vector projection results, and space-curve formulas for differentiation of parametric vectors, arclength, curvature, the TNB frame, and acceleration decomposition.

Oct 13, 2026

Upload by noah