Integration: Inverse Process of Differentiation

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Integration refers to the summation of discrete data. Integration is usually calculated to determine the area, displacement, volume, area, etc that occurs due to small data which cannot be measured individually. The basic idea of limit is where algebra and geometry are implemented. Limits are useful to determine the closeness of two points on graphs until their distance is 0. The fundamental calculus methods of differentiation and integration are employed to solve problems in both mathematics and physics.

Read More: Applications of Derivatives

Table of Content

What is Integration
Types of Integrals
Formulas of Integration
Properties of Integration
Methods of Integration
Important Topics for JEE Main
Things to Remember
Sample Questions

Key Terms: Integration, Definite Integrals, Integrals, Calculus , Indefinite Integrals, Partial Fraction, Algebra, Geometry, Differentiation, Area, Displacement, Volume

What is Integration?

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Integration can be defined as integrating small parts into one whole part. Integration involves finding the antiderivative of a function of f(x). When the function is integrable and its integral is within a finite domain with its limits specified, then it is known as definite integration. The integral is usually denoted by the sign “∫''.

Integration

Integration

The concept of integration is useful for solving problems in which derivatives are given an integral to be calculated. It is also useful for finding out the area of different curves like parabola, ellipse, etc. Integral Calculus consists of Indefinite Integrals and Definite Integrals.

If F(x) is the integral of f(x), then the integral of f(x) is given by,

∫ f(x) dx = F(x) + C

Where,

C → Arbitrary constant.

Variable dx is known as the integrator.

Read More: Methods of Integration

Integrals Detailed Video Explanation

Types of Integrals

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The two types of integrals are described below.

Indefinite Integrals

Integrals that are not defined within specified limits are known as Indefinite Integrals. These integrals don’t contain upper and lower limits.

Indefinite Integral

Indefinite Integral

Definite Integrals

Integrals that can be defined within specified limits are known as Definite Integrals. These integrals contain upper and lower limits.

\(\int_a^b f(x)dx = \lim\limits_{n \to \infty} \sum_{i = 1}^n f(x_i) \Delta x\)

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Relevant Concepts
Definite Integral	Fundamental Theorem of Calculus	Complex Numbers and Quadratic Equations
Application of Integrals	Differential Equations	Integration by Partial Fractions

Formulas for Integration

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In Integration, basic formulas are required to solve the questions. Some of the basic formulas are tabulated below.

S.no	Formula
1	∫ 1 dx = x + C
2	∫ a dx = ax+ C
3	∫ xⁿ.dx = x^{(n + 1)}/(n + 1)+ C
4	∫ sin x dx = – cos x + C
5	∫ cos x dx = sin x + C
6	∫sec²x.dx = tan x + C
7	∫cosec²x.dx = -cot x + C
8	∫ sec x (tan x) dx = sec x + C
9	∫ cosec x ( cot x) dx = – cosec x + C
10	∫ (1/x) dx = log \|x\| + C
11	∫ e^x.dx = e^x + C
12	∫ ax.dx = ax /loga ; a>0, a≠1
13	∫ a dx = ax + C, where a is the constant
14	∫ (1/x) dx = log(x)+ C
15	∫1/(x² - a²⁾.dx = 1/2a.log\|(x - a)(x + a\| + C
16	∫ 1/(a² - x²⁾.dx =1/2a.log\|(a + x)(a - x)\| + C
17	∫1/(x²+ a²).dx = 1/a.tan-1x/a + C
18	∫1/√(x² - a²)dx = log\|x +√(x² - a²)\| + C
19	∫ √(x² - a²).dx =1/2.x.√(x² - a²)-a²/2 log\|x + √(x² - a²)\| + C
20	∫1/√(a² - x²).dx = sin-1 x/a + C
21	∫√(a² - x²).dx = 1/2.x.√(a² - x²).dx + a²/2.sin-1 x/a + C
22	∫1/√(x² + a² ).dx = log\|x + √(x² + a²)\| + C
23	∫ √(x² + a² ).dx =1/2.x.√(x² + a² )+ a²/2 . log\|x + √(x² + a² )\| + C

Properties of Integration

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Basic properties are imperative while solving the questions. Some of the properties of Integration are as follows.

∫ k f(x) dx = k ∫ f(x) dx, where k is any real number.
∫ f(x) dx = ∫ g(x) dx, if ∫ [f(x)-g(x)] dx = 0
∫ [f(x)+g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
∫ [f(x)-g(x)] dx =∫ f(x) dx - ∫ g(x) dx

Some standard function integrals are listed in the table below.

Function	Integral
xⁿ	\(\frac{x ^{n+1}}{n+1}\)
\(\frac{1}{x}\)	ln(x)
e^x	e^x
sin(x)	-cos(x)
cos(x)	sin(x)
sec²(x)	tan(x)

Also Read: Maxima and Minima

Methods of Integration

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The various methods of solving problems using integration are discussed below.

i) Integration by Decomposition

This method involves decomposing function into the sum or difference of many individual integrals. The integral can be algebraic, trigonometric, exponential or a combination of any of these functions. This method is generally used when different functions are added or subtracted.

Example:

Integrate: ∫ cos(x) + sin(x).dx

→ ∫ cos(x) + ∫ sin(x)

By applying formulas of integration

We get,

∫ cos(x) + sin(x).dx = sin(x) - cos(x) + C

ii) Integration by Substitution

This method involves substituting the actual integral with another variable. Generally, this method is used when the derivative of a function is given in the question. It is done to make the calculations easier.

Example:

Integrate ∫ secx tanx / secx .dx

Let, secx = t

so, secx tanx = dt/dx

Secx tanx.dx = dt

→ ∫ dt/t

→ log|t| + C

→ log|secx| + C

iii) Integration by Partial Fraction

This method is used when the integral is in the form of an improper fraction. The fraction is expressed as the sum of a polynomial and proper fraction. The two parts are separately integrated.

Example:

Integrate: ∫ 1 / (x-1) (x+2) (x-3)

∫[ A / (x-1) + B / (x+2) + C / (x-3)

By solving, yields

A = -1/6, B= 1/15, C= 1/10

→ ∫ -1/ 6( x-1) + 1/15( x+2) + 1/ 10(x-3) .dx

→ -log|x-1|/6 + 1 log|x+2|/15 + 1 log|x-3|/10 + C

iv) Integration by Parts

This method is used when two different functions are being multiplied and can't be solved using the aforementioned methods. The formula for this method is -:

∫uv. dx=u∫v

Where,

u → Function of u(x)

dv → Variable dv

v → Function of v(x)

du → Variable du

Example: log x ∫ x.dx – ∫ (\(\frac{d(logx)}{dx} \)) . ∫ x.dx) . dx

= log x \(\frac{x^2}{2}\) – ∫ (\(\frac{1}{x}. \frac{x^2}{2}\)) . dx

= log x .\(\frac{x^2}{2}\) – ∫ \((\frac{x}{2})\) . dx

= log x . \(\frac{x^2}{2}\) – \(\frac{1}{2}. \frac{x^2}{2}\)

= log x . \(\frac{x^2}{2}\) – \(\frac{x^2}{4}\)

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Related Topics
Matrices	Types of Matrices	Operations on Matrices
Transpose of a Matrix	Symmetric and Skew Symmetric Matrix	Invertible Matrix

Important Topics for JEE Main

As per JEE Main 2024 Session 1, important topics included in the chapter integration are as follows:

Formulas of Integration
Properties of Integration
Methods of Integration

Some memory based important questions asked in JEE Main 2024 Session 1 include:

∫0π dx/(1-2a cosx + a²) = ?

Things to Remember

Integration means integrating small parts which can't be measured individually into one whole part.
There are two types of Integrals: Indefinite Integrals and Definite Integrals.
Indefinite integrals are the integrals that are not defined within particular limits, while definite integrals are the integrals that are defined within particular limits.
Integration is also known as anti-derivative because it is the opposite of derivation.
The answer of integration is usually expressed with an arbitrary constant.
The methods of solving problems based on integration are - Integration by Decomposition, Integration by Substitution, Integration by Partial Fraction and Integration by Parts.

Previous Year Questions

The value of sin⁡²51^∘+ sin^⁡239^∘is… [KCET – 2020]
If cos x = |sin x| then, the general solution is… [KCET – 2019]
If 0≤ x< π/2, then the number of values of x… [JEE Main – 2019]
A vertical lamp-post at the midpoint D… [JEE Main – 2019]
If tanA+cotA=2, then the value of… [KCET – 2020]
The value of cos²45^∘−sin²15^∘ is… [KCET – 2017]
A, B and C are the angles opposite to the corresponding sides of lengths… [JKCET – 2017]
The value of tan 8/π is equal to… [KCET – 2016]
The value of tan⁡10^∘tan⁡20^∘tan⁡30^∘tan⁡40^∘tan⁡50^∘tan⁡60^∘… [COMEDK UGET – 2012]
A value of θ satisfying sin⁡5θ−sin⁡3θ+sinθ… [KCET – 2011]

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Sample Questions

Ques. Integrate: xsin³x (5 marks)

Ans. Let I=∫xsin³xdx

Using by parts,

I=x∫sin³xdx−∫{( d/dx)∫sin³xdx}

=x( -3cos³x )−∫1⋅( -3cos³x )dx

= -3xcos³x + 3.1∫cos³xdx

= -3xcos³x + 9 sin³x+C

Ques. Integrate: sin 3x cos 4x (3 marks)

Ans. I=∫sin3xcos4xdx

I= 2.1∫[sin(3x+4x)+sin(3x−4x)]dx

I= 2.1∫(sin7x−sinx)dx

I= 2.1 ( -7cos7x+cosx)+C

I= -cos7x/14 +cosx/2+C

Ques. Integrate: cot x /log sin x (5 marks)

Ans. Take log (sin x) = t

So we get cos x/sin x dx = dt

By cross multiplication cot x dx = dt

By integrating w.r.t. t ∫1/t dt = log t + c

By substituting the value of t, we get

t = log (log sin x)

Ques. Integrate: sinx sin (cos ) (2 marks)

Ans. Put cosx = t ⇒ −sinxdx = dt

⇒∫sinx⋅sin(cosx)dx = −∫sintdt

=−[−cost]+C

=cost+C=cos(cosx)+C

Ques. Integrate: secx(secx.tanx ) dx (2 marks)

Ans. By expanding we get, ∫(sec2x - secx. tanx)dx

By integrating the function we get

= tanx – secx + c

Ques. Integrate sin 2x (3 marks)

Ans. The anti derivative of sin 2x is a function of x whose derivative is sin 2x.

It is known that, \(\frac{d}{dx}\) (cos 2x) = = – 2 sin 2x

⇒ sin 2 x = – \(\frac{1}{2}\)\(\frac{d}{dx}\) (cos 2x)

∴ sin 2x = \(\frac{d}{dx}\)( – \(\frac{1}{2}\) cos 2x)

Therefore, the anti derivative of sin 2x is – \(\frac{1}{2}\)cos 2x.

Ques. Integrate cos 3x (3 marks)

Ans. The anti derivative of cos 3x is a function of x whose derivative is cos 3x.

It is known that, \(\frac{d}{dx}\) = (sin 3x) = 3cos 3x

⇒ cos 3x = \(\frac{1}{3}\)\(\frac{d}{dx}\)(sin 3x)

∴ cos 3x = \(\frac{d}{dx}\)( \(\frac{1}{3}\)sin 3x)

Therefore, the anti derivative of cos 3x is \(\frac{1}{3}\)sin 3x.

Ques. Integrate (ax + c)² (5 marks)

Ans. The antiderivative of (ax + b)² is the function of x whose derivative is (ax + b)²

It is known that,

\(\frac{d}{dx}\) (ax + b)³ = 3a (ax + b)²

⇒ (ax + b)² = \(\frac{1}{3a}\)\(\frac{d}{dx}\)(ax + b)³

∴ (ax + b)² = \(\frac{d}{dx}\) (\(\frac{1}{3a}\)(ax + b)³)

Therefore, the anti derivative of (ax + b)² is \(\frac{1}{3a}\)(ax + b)³.

Ques. Integrate (ax²+ bx + c) dx (3 marks)

Ans. ∫ (ax² + bx + c) dx

= a ∫ x²dx + b ∫ xdx + c ∫ 1 . dx

= a \((\frac{x^3}{3})\) + b \((\frac{x^2}{2})\) + cx + C

= \(\frac{ax^3}{3} + \frac{bx^2}{2}\) + cx + C

Ques. Integrate ∫(2x²+ e^x) dx (2 marks)

Ans. ∫ (2x² + ex) dx

= 2 ∫ x²dx + ∫ e^xdx

= 2 \((\frac{x^3}{3})\) + e^x + C

= \(\frac{2}{3}\)x³ + e^x + C

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Integration: Inverse Process of Differentiation

What is Integration?