Integrals: Overview, Questions, Preparation

In differential calculus, we have to find the derivative or a differential of a given function. However, in integral calculus, one has to find a function whose differential is provided. Integration is the inverse of differentiation. C is the integration constant. Integration is the process that refers to the inverse of differentiation. Considering, ∫f(x) dx = F(x) + C, these types of integrals are known as General Integrals or Indefinite Integrals. Where C is the arbitrary constant that differs with multiple antiderivatives for the given function. 

Note: a derivative of the function is specified, but a function can have many integrals or antiderivatives.

First, let us begin by understanding the symbols and the terms we use in integration.

1. ∫ xndx=xn+1n+1+C
2. ∫ dx=x+C
3. ∫ cosx dx=sinx +C
4. ∫ sinx dx=-cosx +C
5. ∫ sec2x dx=tanx +C
6. ∫ cosec2x dx=-cotx +C
7. ∫ secx tanx dx=secx +C
8. ∫ cosecx cotx dx=-cosecx +C
9. ∫ ex dx=ex +C
10. ∫ 1x dx=logx+C   
11. ∫ ax dx=axloga +C

You can very well be asked to find the value of this constant C. The function and its corresponding x values are given to help you find the value of C. There are two types of integration: definite and indefinite.

Definite integration has defined limits, and hence you obtain a single number or value. It is nothing but the difference between the upper and lower integral values. However, in indefinite integration, you get a function wherein you can substitute numbers to obtain values.

A definite integral can be represented by  ∫_b^a f(x) dx. This is also known as Riemann integral.

Indefinite Integral And Their Properties

Integration is the process that refers to the inverse of differentiation. Considering, ∫f(x) dx = F(x) + C, these types of integrals are known as General Integrals or Indefinite Integrals. Where C is the arbitrary constant that differs with multiple antiderivatives for the given function. 

Note: a derivative of the function is specified, but a function can have many integrals or antiderivatives.

Integration Through Trigonometric Identities

In specific integrals, when the integral involves few or many trigonometric functions, then below-mentioned identities can be used to find the integer.

2 sin A . cos B = sin( A + B) + sin( A – B)

2 cos A . sin B = sin( A + B) – sin( A – B)

cos² A – sin² A = cos 2A

sin² A + cos² A = 1

Methods Of Integration

There are various methods for finding integration in any given function. These methods are-

• By substitutions: This method is used when the variable is substituted by another ideal variable to ease the integration process.
• By parts: If there are any two functions f1 (x), and f2 (x), then presume ∫[f1 (x) f2 (x)] dx = f1 (x) ∫ f2 (x)] dx - ∫ { F1 (x) ∫f2 (x)] dx} dx. 
• By using partial functions: If there are any two polynomials i.e. p1 (x), and p2 (x) on Y, where p2 (x) 0 and degree of polynomial p1 (x)> p2 (x)

Importance and Weightage

In this chapter, students are familiarised with the concept of integration. Students gain an insight into how a function can be found when its derivative is given and learn to derive the area and volume of graphs within specific limits. This chapter is also about definite and indefinite integration and the techniques employed to obtain these integrals.The students can invest some effort, learn the formulas, and conduct a thorough study to make it one of the most interesting topics for examination point of view. Integration is a subsection that comes under the section “calculus”. It carries nominal marks in class X. It carries 5 marks in class XI. It carries 35 marks, in class XII. 

1. Find the anti-derivatives of the following functions.

(i) cos3x

Solution. We have ∫ cos3x = (1/3)sin3x + C, where C is the constant of integration.

(ii) sin2x - 4e3x

Solution. We have ∫sin2x dx -  ∫ 4e3x dx =(-cos2x/2+ C1) - (4/3 e3x +C2 )

= -cos2x/2 - 4/3 e3x + C

Here C =  C1 - C2, which is the constant of integration in this case.

(iii) (2- 3 sinx)/cos²x

Solution. We have  ∫ 2/cos² x dx - 3sinx/cos² x dx

= ∫ 2sec²x dx - ∫3secx tanx dx

= 2tanx - 3secx + C

C is again the constant of integration.

2. Solve ∫ 3ax/(b2 +c2x2) dx

Solution. Let us take v = b2 +c2x2, then

dv = 2c2x dx

Thus, ∫ 3ax/(b2 +c2x2) dx

= (3ax/2c2x)∫dv/v

= (3a/2c2)∫dv/v

= (3a/2c2) log |b2 +c2x2| + C

3. Write the antiderivatives of the function: 3x2+4x3

Solution. ∫3x2+4x3 dx = 3(x3/3) + 4(x4/4)

= x3 + x4

Thus, the antiderivative of 3x2+4x3 = x3 + x4

4. Integrate the function 2x sin(x2+ 1) with respect to X, using substitution method

Solution. Given function: 2x sin(x²+ 1)

Using the substitution method, we get

x² + 1 = t, so that 2x dx = dt.

= ∫ 2x sin ( x² +1) dx = ∫ sint dt

= – cos t + C

= – cos (x² + 1) + C

Where C is an arbitrary constant

Therefore – cos (x² + 1) + C

Q: What is integration?

Q: Explain integral in general terms?

Q: What is the difference between indefinite and definite integral?

Q: What are the differences between differentiation and integration?

Q: What are the similarities between differentiation and integration?

Q: Can we integrate all the functions?

Q: Where do we use integration in our real lives?

Q: How do we find the area of a curve?

Q: What are the various types of integration?