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Integration can be addressed as the reverse process of differentiation. That is why it is also called the 'Inverse Differentiation'. In differential calculus, primary focus is given to rate of change, slope of tangent lines and velocities; but in integral calculus primary focus is given to total size or values, e.g. areas, lengths and volumes. Integral calculus is based on finding the integrals.
Read More: Isosceles Triangle Theorems
| Table of Content |
Key Terms: Integral Calculus, Inverse Differentiation, Derivative, Function, Variable, Integrand, Anti-Derivatives, Integrals, Calculus
Integral Calculus
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Integral calculus is a branch of calculus that deals with the theory and applications of integrals. Integration is the process of finding integrals. The anti-derivatives of a function can be discovered using integral calculus.
Integral calculus deals with total values, such as lengths, areas, and volumes. A few equations with supplied data can have approximations of solutions found using the integral.
There are two types of integration used in integral calculus:
- Indefinite Integrals
- Definite Integrals
Read More:
| Relevant Concepts | ||
|---|---|---|
| Methods of Integration | Integration by Parts | Definite Integral |
| Integration by Partial Fractions | Fundamental Theorem of Calculus | Maxima and Minima |
What are Integral Calculus Formulas?
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The integral for a function at any point on the graph is found using integration. It is an algebraic method. The integral comes from solving the area problem.
Let us assume that F(a) is an anti - derivative of f(a), so F(a) is such that
| F'(a) = f(a) |
The integral of a function f(x) with respect to variable x is given by F(x) and it is represented by,
| ∫ f(x) dx = F(x) + C |
Here,
- F(x) + c = integral of f(x) concerning x.
- x is the variable of integration.
- F(x) is named as anti - derivative or primitive.
- f(x) is called the integrand.
- C is called the constant of integration, and
- dx is called the integrating agent.
Integrals Detailed Video Explanation
Read More: Invertible Matrix
Important Formulas of Integral Calculus
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Some of the important formulas of Integral Calculus:
- ∫ k f(x) dx = k ∫ f(x) dx where k is any number.
- ∫ –f(x) dx = –∫ f(x) dx.
- ∫ f(x) ± g(x) dx = ∫ f(x) dx ± ∫ g(x) dx.
- ∫xn dx = 1 / n+1 x (n+1) + C
unless n = -1
- ∫ ex dx = ex + C
- ∫ 1 / x dx = 1n x + C
- ∫ sinx dx = −cos x + C
- ∫cos x dx = sin x + C
- ∫ sec2 x dx= tan x + C
- ∫ 1 / 1+x2 dx = arctan x + C
- ∫ ax dx = ax/ 1n a + C
- ∫ loga x dx=1/ 1na . 1/x + C
- ∫ 1 / √1−x2 dx = arcsin x + C
- ∫ 1 / x √x2 −1 dx= arcsec x+C
Read More: Differential Equations
Important Terms used in Integral Calculus
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Some of the important phrases used in Integral Calculus are tabulated below:
| Terms | Meaning |
|---|---|
| ∫ f(x) dx | Integral of 'f' with respect to 'x'. |
| f (x) in ∫ f(x) dx | Integrand |
| 'x' in ∫ f(x) dx | Variable of integration |
| An integral of 'f' | A function F such that To (x) = f (x) |
| Integrate | Find the integral |
| Integration | The process of finding the integral |
| Constant of Integration | Any real number C, is considered as constant function |
Read More: Properties of Determinants
Things to Remember
- Integration can be addressed as the reverse process of differentiation.
- Integration is also called the 'Inverse Differentiation'.
- Integral calculus is based on finding the integrals.
- The integral of a function f(x) with respect to variable x is given by F(x) and it is represented by,
- ∫ f(x) dx = F(x) + C
- C is called the constant of integration.
- ∫ k f(x) dx = k ∫ f(x) dx where k is any number.
Read More:
| Related Topics | ||
|---|---|---|
| Matrices | Types of Matrices | Operations on Matrices |
| Isosceles triangle | Scalene Triangle | Types Of Triangles |
Previous Year Questions
- The area enclosed between this curve and the coordinate axes… [JKCET – 2017]
- The area (in sq units) of the region described by… [JEE Main – 2014]
- The area (in s units) of the quadrilateral formed by the tangents… [JEE Main – 2015]
- The area (in s units) bounded by the curve y… [COMEDK UGET – 2013]
- If the line x = b bisects the area bounded by the curves, C1 and C2… [JEE Main – 2020]
- The greatest integer less than or equal to “t”… [JEE Main – 2019]
- [x] denotes the greatest integer less than 20Cr … [JEE Main – 2019]
- The value of the integral… [UPSEE – 2018]
- \(\int {x^3 - 1}\over {x^3 - x}\)dx… [COMEDK UGET – 2015]
- The only integral root of the equation … [AMUEEE – 2016]
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Sample Questions
Ques. Evaluate the following: ∫ 4x2 − 3 × 17√x + 3/x dx. ( 2 marks)
Ans. = ∫ (4x2 − 3x + 3x)dx
= 4x3/3 − 3x2/2 + 3 1n x + C
Here, C is the integral constant.
Ques. Evaluate the following indefinite integral: ∫x3 + 3x – 9dx. ( 2 marks)
Ans. = ∫x4 + 3x – 9dx
=¼ x4 + 3/2 x2 – 9x + C
Here, C is the integral constant.
Ques. Evaluate the following- ∫ (4x2 − 4x) dx. ( 3 marks)
Ans. ∫ (4x2 − 4x) dx
= ∫ 4x2 dx - ∫ 4x dx
= 4 ∫ x2 dx - 4 ∫ x dx
= 4 x3/3 - 4 x2/2 + C
= 2x2 ( 2x/3 - 1) + C
Ques. Evaluate the following- ∫ (4x2 − 2x) dx. ( 5 marks)
Ans. ∫ (4x2 − 2x) dx
= ∫ 4x2 dx - ∫ 2x dx
= 4 ∫ x2 dx - 2 ∫ x dx
= 4 x3/3 - 2 x2/2 + C
= 4 x3/3 - x2 + C
= x2 ( 4x / 3 - 1) + C
Ques. Evaluate the following- ∫ (2x2 − 2x) dx. ( 5 marks)
Ans. ∫ (2x2 − 2x) dx
= ∫ 2x2 dx - ∫ 2x dx
= 2 ∫ x2 dx - 2 ∫ x dx
= 2 x3/3 - 2 x2/2 + C
= 2 x3/3 - x2 + C
= x2 ( 2x / 3 - 1) + C
Ques. Evaluate the following - ∫ √ (4x + 4) ½ dx. ( 2 marks)
Ans. ∫ √ (4x + 4) ½ dx
= (4x + 4) 2 / 4 × 3/2 + C
= (4x + 4) 3/2 / 6 + C
= √(4x + 4)3/ 6 + C
Ques. Evaluate the following- ∫ (2x3 − 4x) dx. ( 5 marks)
Ans. ∫ (2x3 − 4x) dx
= ∫ 2x3 dx - ∫ 4x dx
= 2 ∫ x3 dx - 4 ∫ x dx
= 2 x4/4 - 4 x2/2 + C
= x4/2 - 2x2 + C
= x2 (x2/2 - 2) + C
Ques. Evaluate the following- ∫ (2x4 − 2x) dx. ( 3 marks)
Ans. ∫ (2x4 − 2x) dx
= ∫ 2x4 dx - ∫ 2x dx
= 2 ∫ x4 dx - 2 ∫ x dx
= 2 x5/5 - 2 x2/2 + C
= 2x5/5 - x2 + C
= x2 (2x3/5 - 1) + C
Ques. Evaluate the following- ∫ (2x3 − 2x2) dx. ( 3 marks)
Ans. ∫ (2x3 − 2x2) dx
= ∫ 2x3 dx - ∫ 2x2 dx
= 2 ∫ x3 dx - 2 ∫ x2 dx
= 2 x4/4 - 2 x3/3 + C
= x4/2 - 2x3/3 + C
= x3 (x/2 - ?) + C
Ques. Evaluate the following - ∫ cos a + a d(a). ( 2 marks)
Ans. ∫ cos a + a d(a)
= ∫ cos a da + ∫ a d(a)
= sin a + a2/2 + C
(Evaluating the value of each we got this)
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