This document is a group project submission on integration and trigonometric integrals. It contains the names and student IDs of the five group members. The document defines integration and trigonometric integrals, lists some applications of integration, and discusses special trigonometric integrals involving powers of sine and cosine, tangent and secant, and using integral identities. Examples are provided for evaluating integrals of trigonometric functions like sin5xcos2x and secx using properties of trigonometric functions. The submission thanks the reviewer for their support, which will help boost the group's confidence for future presentations.

1. .

2. Submitted by:
Group leader
Md.Mazharul Islam Id:171-009-041
Associates:
1.Md.Abu Sayed Id:171-035-041
2.Md.Mohidul Islam Id:171-011-041
3.Shahariar Ahamad Id:171-034-041
4.Simon shikdar Id.171-012-041
5.Md.Sarowar islam Id.171-047-o41
Submission Date: 7th August 2017

3. SUBMITTED TO

4. Definition and Application
Integration: The process of finding anti- derivatives is called integration.
Trigonometric Integrals: In mathematics, the trigonometric integrals are a family of
integrals involving trigonometric functions. A number of the basic trigonometric integrals are
discussed at the list of integrals of trigonometric functions.
Application of Integration:
1. Area between two curves. Answer is by integration.
2.Find Distance, Velocity, Acceleration using indefinite integral.
3. Average value of a curve can be calculated using integration.
4.Area under a curve and using integration.
5. Center of Mass
6. Find Kinetic energy; improper integrals
7. Probability
8. Arc Length
19. Surface Area

5. Special TRIGONOMETRIC INTEGRALS

6. Special Trigonometric Integrals
In this section, we will learn:
How to use special trigonometric identities to
integrate certain combinations of trigonometric
functions.
We start with
powers of sine and cosine.

7. SINE & COSINE INTEGRALS
Find ∫ sin5
x cos2
x dx
 We could convert cos2
x to 1 – sin2
x.
 However, we would be left with an expression
in terms of sin x with no extra cos x factor.
Example 1

8. SINE & COSINE INTEGRALS
Instead, we separate a single sine factor
and rewrite the remaining sin4
x factor in
terms of cos x.
So, we have:
5 2 2 2 2
2 2 2
sin cos (sin ) cos sin
(1 cos ) cos sin
=
= −
x x x x x
x x x
Example 1

9. SINE & COSINE INTEGRALS
Substituting u = cos x, we have du = sin x dx.
So,
5 2 2 2 2
2 2 2 2 2 2
3 5 7
2 4 6
3 5 71 2 1
3 5 7
sin cos (sin ) cos sin
(1 cos ) cos sin (1 ) ( )
( 2 ) 2
3 5 7
cos cos cos
=
= − = − −
 
= − − + = − − + + ÷
 
= − + − +
∫ ∫
∫ ∫
∫
x xdx x x x dx
x x x dx u u du
u u u
u u u du C
x x x C
Example 1

10. TANGENT & SECANT INTEGRALS
Evaluate ∫ tan6
x sec4
x dx
 If we separate one sec2
x factor, we can express
the remaining sec2
x factor in terms of tangent
using the identity sec2
x = 1 + tan2
x.
 Then, we can evaluate the integral by substituting
u = tan x so that du = sec2
x dx.
Example 2

11. TANGENT & SECANT INTEGRALS
We have:
6 4 6 2 2
6 2 2
6 2 6 8
7 9
7 91 1
7 9
tan sec tan sec sec
tan (1 tan )sec
(1 ) ( )
7 9
tan tan
x x dx x x x dx
x x xdx
u u du u u du
u u
C
x x C
=
= +
= + = +
= + +
= + +
∫ ∫
∫
∫ ∫
Example 2

12. TANGENT & SECANT INTEGRALS
Find ∫ sec x dx
First, we multiply numerator and denominator
by sec x + tan x:
2
sec tan
sec sec
sec tan
sec sec tan
sec tan
x x
xdx x dx
x x
x x x
dx
x x
+
=
+
+
=
+
∫ ∫
∫

13. TANGENT & SECANT INTEGRALS
If we substitute u = sec x + tan x,
then du = (sec x tan x + sec2
x).
 The integral becomes: ∫ (1/u) du = ln |u| + C

14. TANGENT & SECANT INTEGRALS
Thus, we have:
sec ln | sec tan |xdx x x C= + +∫

15. Find
Put,
z= sin x
dz= cos x dx
xdx∫
5
cos
SPECIAL TRIGONOMETRIC INTEGRALS Example 4

16. TRIGONOMETRIC INTEGRALS
Find ∫ sin4x cos5x dx
Use this formula-
Integral Identity
a ∫ sin m
x cos n
x dx
b ∫ sin m
x sin n
x dx
c ∫ cos m
x cos n
x dx
[ ]1
2
sin cos
sin( ) sin( )
A B
A B A B= − + +
[ ]1
2
sin sin
cos( ) cos( )
A B
A B A B= − − +
[ ]1
2
cos cos
cos( ) cos( )
A B
A B A B= − + +

17. TRIGONOMETRIC INTEGRALS
Evaluate ∫ sin 4x cos 5x dx
 This could be evaluated using integration by parts.
 It’s easier to use the identity in Equation 2(a):
Example 5
[ ]1
2
1
2
1 1
2 9
sin 4 cos5 sin( ) sin9
( sin sin9 )
(cos cos9 )
x x dx x x
x x dx
x x C
= − +
= − +
= − +
∫ ∫
∫

18. Thank you sir for supporting us .This
presentation will help us a lot . It will make
our confidence high for further presentation.
THE END