Integrals
In this chapter, the methods to determine the function when its derivative is given and the area under a graph of a function are discussed. The basic properties of integrals and the fundamental theorem of calculus are also included in this chapter.
Integration is the inverse process of differentiation. In differential calculus, we are given a function and we have to find the derivative or differential of this function, but in integral calculus, we are to find a function whose differential is given. Thus, integration is a process that is the inverse of differentiation.
Then, ∫f(x) dx = F(x) + C, these integrals are called indefinite integrals or general integrals. C is an arbitrary constant by varying which one gets different anti-derivatives of the given function.
Derivative of a function is unique but a function can have infinite anti-derivatives or integrals.
Properties of Indefinite Integral
Various properties of the integral are,
- ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx
For any real number k, ∫k f(x) dx = k∫f(x)dx.
In general, if f1, f2,………, fn are functions and k1, k2,…, kn are real numbers, then
- ∫[k1f1(x) + k2 f2(x)+…+ knfn(x)] dx = k1 ∫f1(x) dx + k2 ∫ f2(x) dx+…+ kn ∫fn(x) dx
First Fundamental Theorem of Integral Calculus
Let the area function be defined as
A(x) = ∫axf(x)dx for all x ≥ a
where the function f is assumed to be continuous on [a, b]
Then A’ (x) = f (x) for every x ∈ [a, b].
Second Fundamental Theorem of Integral Calculus
Let f be the certain continuous function of x defined on the closed interval [a, b] then,
∫baf(x)dx = [F(x) + C]ba = F(b)−F(a)
Learn more about, Fundamental Theorem of Calculus
Standard Integrals Formulas
Standard Integral Formulas are,
- ∫xndx = xn+1/(n+1) + C (where, n ≠ −1)
- ∫cos x dx = sin x + C
- ∫sin x dx = −cos x + C
- ∫sec2x dx = tan x + C
- ∫cosec2x dx = −cot x + C
- ∫sec x.tan x dx = sec x + C
- ∫cosec x.cot x dx = −cosec x + C
- ∫exdx = ex + C
- ∫axdx = axlogea + C
- ∫1/x dx = log|x| + C
Other Integral Formulas
Other integral formulas that are widely used are,
- ∫tan x dx = log|sec x| + C
- ∫cot x dx = log|sin x| + C
- ∫sec x dx =log|sec x + tan x| + C
- ∫cosec x dx = log|cosec x − cot x| + C
Class 12 Maths Formulas
Class 12 maths formulas page is designed for the convenience of the learners so that one can understand all the important concepts of Class 12 Mathematics directly and easily. Math formulae for Class 12 are for the students who find mathematics to be a nightmare and difficult to grasp. They may become hesitant and lose interest in studies as a result of this. We have included all of the key formulae for the 12th standard Maths topic, which students may simply recall, to assist them in understanding Maths in a straightforward manner. For all courses such as Integration, Differentiation, Trignometry, Relation and Functions, and so on, the formulae are provided here according to the NCERT curriculum.
Table of Content
- Chapter 1: Relations and Functions
- Chapter 2: Inverse Trigonometric Functions
- Chapter 3: Matrices
- Chapter 4: Determinants
- Chapter 5: Continuity and Differentiability
- Chapter 6: Applications of Derivatives
- Chapter 7: Integrals
- Chapter 8: Applications of Integrals
- Chapter 9: Differential Equations
- Chapter 10: Vector Algebra
- Chapter 11: Three-Dimensional Geometry
- Chapter 12: Linear Programming
- Chapter 13: Probability
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