Calculus formula integration,differentiation,limits&derivatives, Cheat Sheet of Mathematics

Download Calculus formula integration,differentiation,limits&derivatives and more Cheat Sheet Mathematics in PDF only on Docsity! C ALCULUS FORMULA & IDENTITIES. Calculus is one of the branches of Mathematics that is involved in the study of ‘Rate of Change’ and their application to solving equations. It has two major branches, Differential Calculus that is concerning rates of change and slopes of curves, and Integral Calculus concerning accumulation of quantities and the areas under and between curves. Both branches make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit. These two branches are related to each other by the fundamental theorem of calculus. DIFFERENTIAL CALCULUS: The Differential Calculus splits up an area into small parts to calculate the rate of change. The Integral calculus joins small parts to calculates the area or volume and in short, is the method of reasoning or calculation. In this page, you can see a list of Calculus Formulas such as integral formula, derivative formula, limits formula etc. Since calculus plays an important role to get the optimal solution, it involves lots of calculus formulas concerned with the study of the rate of change of quantities. It describes the rate of change of a function for the given input value using the derivative of a function. The process of finding the derivative of a function is called 1 differentiation. The reverse process of the differentiation is called integration. This process helps to maximize or minimize the function for some set, it often represents the different range of choices for some specific conditions. The function allows us to compare the different choices where it uses different calculus formulas to chooses the best optimal solution. Differential Calculus Formulas: Differentiation is a process of finding the derivative of a function. The derivative of a function is defined as y = f(x) of a variable x, which is the measure of the rate of change of a variable y changes with respect to the change of variable x. It means that the derivative of a function with respect to the variable x. There are a number of rules to find the derivative of a function. These rules make the differentiation process easier for different functions such as trigonometric functions, logarithmic functions, etc. Here, a list of differential calculus formulas is given below: 2 Sectional Method Formula / Method of Parts formula: Generally, the correct choice of u and v is to be done to find the integration by the block method. Therefore, a formula works in relation to this, which is represented by the name of ILATE. Its meaning is defined as follows. • I = Inverse Trigonometry Function • L = Logarithm Function • A= Algebraic Function • T = Trigonometry Function • E = Exponential Function Integration Formula List: Integration is the most important topic in Calculus, so, it is our responsibility to get specific information about it so that it is easy to solve the question. Here the list of all the formulas is provided in a systematic manner which is essential for class 12. • ∫1 dx = x + C • ∫ a dx = ax+ C • ∫ (1/x) dx = ln |x| + C • ∫ ex dx = ex+ C • ∫ sin x dx = – cos x + C • ∫ cos x dx = sin x + C 5 • ∫ sec2x dx = tan x + C • ∫ csc2x dx = -cot x + C • ∫ sec x (tan x) dx = sec x + C • ∫ csc x ( cot x) dx = – csc x + C • ∫cosec2x.dx = -cotx + C • ∫secx.tanx.dx = secx + C • ∫cosecx.cotx.dx = -cosecx + C • ∫tanx.dx =log|secx| + C • ∫cotx.dx = log|sinx| + C • ∫secx.dx = log|secx + tanx| + C • ∫cosecx.dx = log|cosecx - cotx| + C • ∫ ax dx = (ax/ln a) + C ; a>0,  a≠1 Basic integration formula: • ∫ xn.dx = x(n + 1)/(n + 1)+ C • ∫1.dx = x + C • ∫ ex.dx = ex + C • ∫1/x.dx = log|x| + C • ∫ ax.dx = ax /loga+ C • ∫ ex[f(x) + f'(x)].dx = ex.f(x) + C Integration Formulas of Inverse Trigonometric functions: • ∫1/√(1 - x2).dx = sin-1x + C • ∫ /1(1 - x2).dx = -cos-1x + C • ∫1/(1 + x2).dx = tan-1x + C • ∫ 1/(1 +x2 ).dx = -cot-1x + C • ∫ 1/x√(x2 - 1).dx = sec-1x + C 6 • ∫ 1/x√(x2 - 1).dx = -cosec-1 x + C Advanced Integration Formulas: • • ∫1/(x2 - a2).dx = 1/2a.log|(x - a)(x + a| + C • ∫ 1/(a2 - x2).dx =1/2a.log|(a + x)(a - x)| + C • ∫1/(x2 + a2).dx = 1/a.tan-1x/a + C • ∫1/√(x2 - a2)dx = log|x +√(x2 - a2)| + C • ∫ √(x2 - a2).dx =1/2.x.√(x2 - a2)-a2/2 log|x + √(x2 - a2)| + C • ∫1/√(a2 - x2).dx = sin-1 x/a + C • ∫√(a2 - x2).dx = 1/2.x.√(a2 - x2).dx + a2/2.sin-1 x/a + C • ∫1/√(x2 + a2 ).dx = log|x + √(x2 + a2)| + C • ∫ √(x2 + a2 ).dx =1/2.x.√(x2 + a2 )+ a2/2 . log|x + √(x2 + a2 )| + C sin2 x ( 1 – cos 2x ) / 2 cos2 x ( 1 + cos 2x ) / 2 sin3 x ( 3 sin x – sin 3x ) / 4 cos3 x ( 3 cos x + cos 3x ) / 4 tan2 x sec2 x – 1 sin2 x + cos2 x 1 tan2 x cosec2 x – 1 7 1 / sin” (x) dx = r / cos(a) de = + / tan” (a) de = ro - / — dz / cot”(2) da = ——* cot"*(2) — / cot"-?(2) de / see"(a) dr = —*— sec"? (2) tan(2) + 2" = / sec" (x) dn [oser@) dz = sin” ~*(x) cos(x) + <_ [se@) dar 3"? (0) daz 7 en 2 (x) cot(z) PB Posed Caption Ta = dx = sin! r+C -f : dz =tan124+C +z? ‘Swe 5 dt = sec? z+C + fsin"(x)dz = —sin” 1 (a) cos(x) + "— fsin” ?(x)dx + fcos"(x)dx = +cos”-!(x) sin(x) + 2 fcos”-?(x)dx + ftan"(x)dr = —itan” 1x) — ftan”?(x)da + fsec”(x)dz = =tysec”? (2) tan(x) + acd + fsec”?(a)dx + fese"(x)da = —Lesc"? (x) cot(x) + 2 esc”? (x)dx Caption 10 Formula of Partial Fractions 
 11 Caption 12 sin'xdx=xsin' x+V1-x° +C tan”'x dx = xtan™' x— pint 2°) +C sec 'xdx=xsec' x—Inx+ Vx°-1+C 1 . xsin'x dx = gee —Dsin xt? VI-x+C r -1 1 -l x xtan tar= 5G +1) tan z—are r x 1; xsec 'xdx= "sec! x— 5 Vx' -1+C e x 1 n+l x" sin'x dx = >— sin x-—— | ~—— de +Cifn#-1 n+l n+1J j-x? e n+l nel x" tan'x dx =~ — tan x- | * 7 ax+Cifn#-l n+1 n+1J1+x ° n+l 1 n x" sec 'x dx = ~—sec x-—— | —*— de + Cifn#-1 n+1 nt+lJd Jy?-] Caption