“Discover integration in calculus, the process of finding the area under curves, summing infinitesimal changes, and solving problems related to accumulation and total quantities.”
Mathematics Class 12 Chapter 3 Notes
? Cos(ax+b)dx=: (ax+b)n+1/a(n+1) +c
The differentiate of x is given by: sin-1 (x/a) + c
? Sin(ax+b)dx=: y= -Cos x + c
If y= x2-1 then dy =: 2x dx
?x Sin x dx=: 1/a Sec(ax+b)+c
d/dx ?…… dx are: 1/n+1 xn+1 + c
The solution of dy/dx = -y is: y=ce-x
The solution of xdy/dx = 1 + y is: 1/a ln(ax+b)+c
? Cosx /Sinx ln Sinx dx =: x Sin-1 x + ?(1-x2) + c ln(ln Sin x) + c
?3-1 x3 dx =: 20
?etan-1/1-x2 dx =: etan-1 x + c
?eax[af(x)+f'(x)]dx=: eax f(x) + c
-? ecotx-1/1+x2 dx =: ecot-1x + c
Substitution for the integrals involving the expression ?a2-x can be: x = a sin?
The solution of differential equation dy/dx = 1/?x2-1 is: y = cosh-1x + c
? Cosec2 (ax+b)dx=: x lnx – x + c
? (ax+b)n dx= _______ if n? -1 then: lnx + c
?Sec4x dx =: Sec3x/3 + c
?x exdx=: e2xCos x + c
? exx dx =: xex – ex + c
? xn dx = ____________, if n ? -1 then: dy=f'(x)dx
?e2x(-Sinx + 2Cosx)dx=: Cos-1 (x/a) +c
?x Cos x dx=: 1/a Sin(ax+b) + c
_____ is inverse process of differentiation: integration
? 1/ax+b dx equals: c?a f(x) dx + b?c f(x) dx
An equation containing at least one derivative of a dependent variable with respect to an independent variable is called: Differential equation
? Sec2(ax+b)dx=: dx
? Cosec2(ax +b) d: -1/a Cot (ax+b)+c
? (3x+4)5dx =: (3x+4)6/ 18 + c
?x-1 dx =: xex – ex + c
?-dx/?a2-x2: -1/a Cot (ax+b) + c
??-? 1/1+x2 dx =: ?
?Sin-1x dx=: ex lnx + c
?ex(Sinx + Cosx)dx=: -2
f(x)g(x): f(x)g(x) –
If a < c < b then b?a f(x) dx =: y = Tan(ex+ c)
5?2 f(x)dx = 8, and 1?2 f(x)dx = 3, then 5?1 f(x)dx =: 5
? Cosx [lnSinx /Sinx] dx =: x Sin x – Cosx + c
If y= x2+2xthen dy =: 2(x+1)
If: Integral
? ebx+kdx=: 1/blna abx+k + c
? Cosec(ax+b)Cot(ax+b)dx=: -1/a Cosec(ax+b) + c
? Cot(ax+b)dx=: 1/a ln |Sin(ax+b)| + c
if y = f(x) be different function then: ay = f(x +?) – f(x)
The solution of dy/dx = Sin x is: ex Cox + c
If 2?1 (3x2 +2x – k) dx = 12 then k =: y = ce-x
? Sec x dx=: Sin x – x Cos x + c
1/b ln|ax-b| + c: 3
?ex [Cos x -sin x] dx =: 1/a eax+b+ c
?lnx dx=: ln |Sec x+ Tan x| + c
? Tan(ax+b)dx=: 1/a ln |Sec(ax+b)| + c
?sec2 x/ tan dx + ?cosec2x/cotx dx =: 2 in tan x + c
The conditions through which the orbitrary constants involving in the differential equation can be determined is called: 1/2 (ln Sin x)2+ c
? ax dx =: ax/lna + c
? Sec(ax+b)Tan(ax+b)dx=: -1/a Cos(ax+b)+c
?ex[lnx+1/x]dx=: Initial value conditions
The solution of dy / y2 + 1 = ex dx is: 1/a Tan (ax+b)+c
The differentiate of y is given by: y= cx + 1
? inx dxis equals to: %lnx -x + c
The solution of the differential equation dy/dx + y =0 is: y= Sin-1x + c
The solution of dy = dx /?(: Inverse operations
?ex(Sinx + Cosx)dx=: ex Sin x + c
? 1/?a2 -x2 dx =: 1/a ln|ax+b| + c
? Cosec x dx=: ln |Cosec x – Cot x| + c
If the lower limit is a constant and the upper limit is a variable, then the integral is a function of: upper limit
The term dy (or df) = f ‘(x) dx is called the __________ of the dependent variable y: Differential
f(x) = x cot x is: Even function
lim Sin2x/x =: 0
Which one is an identity function: f(x) = x
Explicit form of y – x2 – 2x + 1 = 0 is: y=x2 + 2x-1
If f : X: Range of f
Let f(x) = 4 – x, g(x) = -2, then fog (x) =: 6
If F(x) = x contx than f(x) is: even function
If f(-x) = -f(x) for all x in the domain of f, then f is: Odd function
The equation of the type x=f(t) and y=g(t) are called: Reciprocal equations
f(x) = cos x + sin x is _____ function: Nor the even nor odd
If y=f(x) then x is called: Independent Variable
If x=ay then y=loga x and a: Natural Logarithm
A function defined by f(x) = x2is: Even function
Which one is not an exponential function: xn
Parametric equations x = a cost t, y = a sin t represent the equation of: Circle
Algebraic functions are those functions, which are defined by _____ expressions: Algebraic
X2 + Y2 = 16 is: Not function
The term function was introduced by: Leinbniz
Which one is an explicit function: y = f(x)
Iff: X: y=f(x)
f : x –> ?3x2-1 and 9 : x –> Sinx , then g of : x –>: sin ?3x2 – 1
For any set X, a function 1 : X: An identity function
The volume V of a cube as a function of the area Aof its base is: V=A3/2
In y = f(x), the variable y is called _____ variable of a function f: Dependent
If y=f(x) then y is called: Dependent Variable
f(x) is even function, if and only if: f(-x) = -f(x)
A function from Y to X is denoted as: f : Y
In binary relation, the set consisting of all the first elements of the ordered pairs is called: Range
cos h2x – sin h2x =: 1
The function f(x) = ax + b is an identity function if: a= 1, b = 0
Let ?(x) = ?x2-9 then range of fis: [0,?]
A function f from a set X to a set Y, then set of corresponding elements y in Y is called the __________ of f: Range
lim sin7?/? =: 7
Limt [3x+4/x+3]: 2
Important : Review all previous years’ 12 Class Past Papers for comprehensive practice and better exam preparation.