This presentation covers the basics of Integration along with appropriate examples.

1. A course on INTEGRAL
CALCULUS
Presented by Gaurav Saha
Targeted for JEE Aspirants and also for those appearing for Class XII board
exams

2. Definition of Integration
▪ The reverse process of differentiation is defined simply as
Integration.
If 𝐹′(𝑥 )= 𝑓(𝑥) , then anti-derivative of 𝑓 (𝑥) is defined as the
Indefinite Integral (or primitive function) of 𝑓 (𝑥) and is
mathematically expressed as: int 𝑓 (𝑥) 𝑑𝑥 = 𝐹 (𝑥) + 𝑐
𝑓(𝑥) is called the integrand and 𝑐 is the constant of integration.
Sign of Integration is

3. Types of Integral
▪ Indefinite Integral :Where the upper and lower limits are not
specified.
▪ Definite Integral: Where the upper and lower limits are specified

4. Graphical Representation of Integration

5. General rules of Integration

6. Integrals of some simple functions

7. Problem 1
▪ Find the indefinite integral for 𝑓(𝑥) = 𝑥^4
Solution: we can write the answer generally
Int (𝑥^4) 𝑑𝑥 = (𝑥^5 )/5 + 𝑐 ( constant)

8. Problem 2: Find the integral of ( 4x^10 –
2x^4 + 15x^2 )/x^3

9. Integrals of trigonometric functions

10. Derivation of few Integrals
▪ Integral of (tan x)

11. Derivation of Few Integrals
▪ Integral of (sec x)

12. Problem 3: Find the integral of (cos
x)^2
▪ Solution:

13. Problem 4: Find the integral of (cos x)^4

14. Methods of Integration
▪ Substitution method: If the integrand is in the form of 𝑓(𝑔 (𝑥 )). 𝑔′(𝑥),
and we substitute 𝑢 = 𝑔(𝑥) we will have:
𝑓 (𝑔 (𝑥)) . 𝑔′ (𝑥) 𝑑𝑥 = 𝑓( 𝑢) . 𝑢′ 𝑑𝑥
= 𝑓 (𝑢) 𝑑𝑢
Now if int 𝑓 (𝑢) 𝑑𝑢 = 𝐹 (𝑢) + 𝑐, then integral of [𝑓 (𝑔( 𝑥 )). 𝑔′ (𝑥) 𝑑𝑥] = 𝐹 (
𝑔 (𝑥)) + 𝑐

15. Problem 5: Find the integral of
x/{(x+1)^1/2}

16. Problem 6: Find the integral of
x(x+1)^1/2

17. Problem 7: Find the integral of [{(sin
x)^6}*(1-(sin x)^2)*cos x]dx

18. Problem 8: Find the integral of [(sin
x)^2 * (cos x)^2]

19. Some important problems
Find the integrals of
▪ (x^7)/(x-1)
▪ (tan x) * (tan 2x )* (tan 3x)
▪ (tan x)/a+{b(tan x)^2}
▪ (x * e^x)/(x+1)^2
▪ (sin x)/(1+sin x)^1/2

20. Some standard integrals by Substitution
method

21. Shortcuts to find integrals of some
special forms
▪ Integral of the form dx/(lx+m)(ax+b)^1/2
Here, we have to put (ax+b)=z^2
▪ Integral of the form dx/(lx+m)(ax^2+bx+c)^1/2
Here we have to put (lx+m)=1/z
▪ For the integral of form dx/(lx^2+m)(ax^2+b)^1/2
Here, we have to put (ax^2+b)^1/2 = xz
▪ For the integral of form dx/[(x-a)^m * (x-b)^n]
Here, we put (x-a)=z(x-b)

22. Hints to some important problems
▪ Find the integral of dx/(2x^2 + 3x + 4)^1/2
HINT: We write (2x^2 + 3x + 4) = 2(x^2 + 3/2 x +2) = 2[(x+3/4)^2 + 23/16]
After substituting this expression in the denominator of the problem, use the
value of standard integral of
dx/(x^2 + a^2)^1/2 = log |x+ (x^2+a^2)^1/2|

23. Problem 10: Find the integral of dx/[(x-
a)(b-x)]^1/2 , a<x<b
▪ HINT: Put (x-a)= z^2
So, dx = 2z dz and x = z^2 + a
Use the value of standard integral of
dx/(a^2 – x^2)^1/2 = (sin x/a)^-1 + c
Substituting these value, find the value of the integral

24. Problems to be solved
▪ Find the integral of
▪ [( x + 1 ) / ( 4 + 8x – 5x^2 )] dx
▪ dx / (x – 3 )*(2 x^2 – 12x + 17) ^1/2
▪ dx / (2x^2 + a^2)*(x^2 + a^2)
▪ dx / ( 2x^2 + 6 – 5x ) ^ 3/2

25. Method of integration
▪ Integration by Parts:

26. Some standard integrals

27. Problem 11: Find the integral of (x * sin 2x)

28. Problem 12: Find the integral of [x * e^(-2x)]

29. Some important problems
Find the integral of
▪ [(x^4)*(log x)^2 ] dx
▪ (sin x)^-1 dx
▪ [cos (x^1/2)]^-1 dx
▪ (e^2x * cos 3x) dx
▪ cos(log x) dx

30. Definite Integral
▪ When the upper and lower limits of an integral are specified
▪ It is applied to find the area under curves.

31. Properties of Definite Integral

32. Solution to some problems

33. Solution to some problems

34. Solution to some problems

35. Important problems from this part
Find the value of integral of
▪ log(1+ tanx ) dx from 0 to π/4
▪ [(1 / log x ) – (1/ log x^2)] dx from 2 to e
▪ dx / {x * (1+log x) ^ 2} from 1 to e
▪ [(cos 2x) * log(sinx )] dx from π/4 to π/2

36. THANK YOU