CBSE Class 12 Mathematics Notes Chapter 7 Integrals

Integrals are a branch of calculus that is used to determine the anti-derivatives of a function. It is used for the representation of the area of a region under a curve. 

• Integrals, also known as antiderivative and primitive functions, are the inverse process of differentiation. 
• It is used to represent the upper and lower limits where the value of x is restricted to lie on a real line.
• The definite integral of a function determines the area of the region bounded by its graph of the given function between two points in the line.
• F(x) is denoted as Newton-Leibniz Integral where every value of x in I, F'(x) = f(x).
• It is used to find lengths, areas, volumes, and the derivation of the antiderivative formula.
• The concept is used for solving displacement and motion problems and kinetic energy and center of mass problems.

According to the CBSE Syllabus 2023-24, the chapter on Integrals comes under Unit 3 of Calculus. NCERT Class 12 Mathematics Unit Calculus holds a weightage of around 35 marks and includes Integrals topics. 

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Class 12 Mathematics Chapter 7 Notes – Integrals

Integration as an Inverse Process of Differentiation

• Integration refers to the process of computing a definite integral and an indefinite integral. 

• Differentiation is used to determine the derivative of a function, whereas integration is used to determine the antiderivative of a function.
• We will calculate the value of the original or primitive function when the derivative of a function is given.
• Mathematically, it can be represented as:

 ∫ f(x) dx = F(x) + C

• Where ∫ f(x) dx represents the indefinite integral of the function f with respect to the variable x.

Some important formulas of Integrals as an Inverse Process of Differentiation are as follows:

• ∫ xⁿ dx=xⁿ⁺¹ /n+1+C, n ≠ -1
• ∫ dx = x + C
• ∫ cosx dx = sinx + C
• ∫ sinx dx = -cosx + C
• ∫ sec²x dx = tanx + C
• ∫ cosec²x dx = -cotx + C
• ∫ sec²x dx = tanx+C
• ∫ secx tanxdx = secx + C
• ∫ cscx cotx dx = -cscx + C
• ∫1/(√(1-x²)) = sin^-1 x + C
• ∫-1/(√(1-x²)) = cos^-1 x + C
• ∫1/(1+x²)= tan^-1 x + C
• ∫-1/(1+x²)= cot^-1 x + C
• ∫1/(x√(x² -1)) = sec^-1 x + C
• ∫-1/(x√(x² -1)) = cosec^-1 x + C
• ∫ e^xdx = e^x + C
• ∫dx/x = ln|x| + C
• ∫ a^x dx = a^x/ln a + C

Integration as an Inverse Process of Differentiation

Methods of Finding Integrals of Functions

There are three methods used for finding integrals of functions, which are as follows:

Integration by Substitution Method

• The substitution method is used when the algebraic function is not given in the standard form.
• The function is reduced in standard form by the process of substitution.
• If u is given as a function of x, then u' = du/dx.

∫ f(u)u' dx = ∫ f(u)du

Steps to calculate Integration of a function by Substitution

• First, determine the variable that needs to be reduced.
• Next, calculate the value of dx of the given integral, where f(x) is integrated with respect to x.
• Convert the substitution function in the form of dx.
• Now, integrate the function obtained in the above step.
• Substitute the initial value of x to obtain the required answer.

Substitution Method

Integration by Parts Method

• The integration by parts method is used to integrate the product of two or more functions. 
• The first function, f(x), is obtained by the derivative formula.
• The second function, g(x), is obtained by the integral of a function. 
• It is also known as the product rule of integration.

∫f(x)g(x) dx = f(x)∫ g(x) dx - ∫ (f'(x) ∫g(x) dx) dx

Integration by Parts Method

Integration by Partial Fractions

• Integration by partial functions involves decomposing the proper rational fraction into a sum of simpler rational fractions. 
• It is used to determine the factor of the denominator and then decompose it into two different fractions.
• Partial Fraction decomposition refers to the decomposition of rational fractions into simpler rational fractions. 

∫f(x)/g(x) dx = ∫ p(x)/q(x) + ∫ r(x)/s(x)

Integration by Partial Fractions Formula

The integration by partial fractions formula are as follows:

Integration by Partial Fractions Formula

Types of Integrals

Integrals are divided into two categories, which are as follows:

Definite Integrals

• Definite integral is a type of integral which involves having a pre-existing value of limits, which in turn makes the final value of an integral definite.
• It is also known as Riemann Integral when the value of constraint lies on a real line.
• Mathematically, it can be represented as:

_a∫^b f(x) dx = F(b) – F(a)

• Where ∫ = Integration symbol
• a = Lower limit
• b = Upper limit
• f(x) = Integrand
• dx = Integrating agent

Definite Integrals

Indefinite Integrals

• Indefinite integral is a type of integral that does not have a pre-existing value of limits, which, in turn, makes the final value of an integral indefinite.
• It returns an independent variable and does not have any upper and lower limits.
• Mathematically, it can be represented as:

∫f(x) dx = F(x) + C

• Where F(x): Antiderivative or Primitive
• f(x): Integrand
• dx: Integrating Agent
• x: Variable of Integration
• C: Constant of Integration

Indefinite Integrals

Fundamental Theorem of Calculus

• The fundamental theorem of calculus establishes the relationship between the differentiation and integration of a function.
• The concept is based on area problems and tangent problems.
• It determines the difference between the antiderivative at the upper and lower limits of the integration process. 

Area Function

• The area function is used to find an area enclosed by a curve.
• Consider a function f(x), which is defined in the interval [a, b].
• The integral of the f(x) is the area enclosed by the curve y = f(x) and the lines x = a, x =b and x–axis a∫x f(x) dx.
• It is assumed that f(x) > 0 and x belongs to (a, b) and is non-negative. 
• Mathematically, it is represented by : 

F(x) = ^x∫_a F(t)dt

Area Function

First Fundamental Theorem of Integral Calculus

• The First Fundamental Theorem states that f is a continuous function on the closed interval [a, b], and A(x) be the area function.

• In such cases, A′(x) = f(x), for all x ∈ [a, b].
• It can be represented as:

A(x)=^b∫_af(x)dx for all x≥a

First Fundamental Theorem of Integral Calculus

Second Fundamental Theorem of Integral Calculus

• The statement of the second fundamental theorem states that if the required function is continuous on the closed interval [a, b], and F is an indefinite integral of a function the given function on intervals [a, b], then the second fundamental theorem of calculus is given as:

F(b)- F(a) = _a∫^b f(x) dx

Second Fundamental Theorem of Integral Calculus

Properties of Definite Integrals

The important properties of definite integrals are as follows:

• ∫_a^b f(x) dx = ∫_a^b  f(t) d(t)
• ∫_a^b f(x) dx = – ∫_b^a f(x) dx
• ∫_a^a f(x) dx = 0
• ∫_a^b f(x) dx = ∫_a^c f(x) dx + ∫_c^b f(x) dx
• ∫_a^b f(x) dx = ∫_a^b f(a + b – x) dx
• ∫₀^a f(x) dx = f(a – x) dx

Properties of Definite Integrals

There are Some important List Of Top Mathematics Questions On Integrals Asked In CBSE CLASS XII

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