This document outlines techniques for integrating irrational and trigonometric functions. It discusses using trigonometric substitutions such as x=atanθ, x=asinθ, and x=asecθ for integrals involving radicals, and substitutions like x=asinh z or x=acosh z for certain radicals. Examples are provided. Exercises involve integrating functions with radicals like √x^2+a^2 and trigonometric functions like sin^5x using reduction formulas and trigonometric identities. Reduction formulas for integrals of trigonometric functions like tanx, secx, and cotx are also explored.

1. TECHNIQUES OF INTERGRATION

2. OUTLINE:
• Integration of Irrational Functions:
 Introduction
 Examples
 Exercise 4.5
• Integration of Trigonometric Functions:
 Introduction
 Examples
 Exercise 4.6

3. Integration of Irrational Functions:

4. INTRODUCTION
The following trignometric substitutions will be used to handle
integers involving the indicated radicals:
• √x2 + a2 , put x=atanθ
• √ a2 - x2 , put x=asinθ
• √x2 - a2 , put x=asecθ
In case where the above substitutions lead to complicated
integrals, it is sometimes convenient to make the hyperbolic
substitutions.
• x=asinh z for integrals involving √x2 + a2
• x=acosh z for integrals involving √x2 - a2

5. Example:01

6. Example:02

7. Exercise 4.5:
Integrate each of the following with respect to
(i ) ∫ x2 √25 - x2

8. (iii) ∫ex √1 - e2x

9. (vii) √x2 + 2x + 3

10. (x) x +1 /√x2 + 2x + 3

11. (xxiii) 1/ (1 - 2x)√1 + 4x

12. (xxiii) x1/ (1 + x2)√1 - x2

13. (xxiv) 1/ (1 - 2x2)√ 1 - x2

14. (xxvii) x3/ √ 1 + x2

15. Integration of Trigonometric Functions:

16. INTRODUCTION
We can use
substitution and
trigonometric identities
to find integral of
certain types of
trigonometric
functions.We begin by
giving the
antiderivatives of the
six basic trigonometric
functions:

17. Example 01:
In this section , will find the Reduction Formula for these intergral
Let we have I = ∫ sinn x dx
I = ∫sin x sinn-1 x dx
By using Integration of parts we have,

18. Examples 02:
Evalute I = ∫ cos5 x dx

19. Find the the reduction formula of these.
(v)∫ tann x dx

20. (vi)∫secnx

21. EXERCISE 4.6:
Integrarate with respect to x.
i.Sin5x

22. (ii)cos7x

23. Find the the reduction formula of these.
(vii) ∫cotnx dx

24. (viii) ∫ cosecn x dx

25. (xiii) 1
a+bsinx

26. (xv) cotx
1 + sinx

27. (xvii) cosx
2 - cosx

28. (xix) 1
4sinx – 3cosx

29. 26(b) ∫cscdx = 1 1-cosx
2 1+cosx