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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 ∫ sin 5 x cos 2 x dx  We could convert cos 2 x to 1 – sin 2 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 sin 4 x factor in terms of cos x. So, we have: Example 1

9. SINE & COSINE INTEGRALS Substituting u = cos x, we have du = sin x dx. So, Example 1

10. TANGENT & SECANT INTEGRALS Evaluate ∫ tan 6 x sec 4 x dx  If we separate one sec 2 x factor, we can express the remaining sec 2 x factor in terms of tangent using the identity sec 2 x = 1 + tan 2 x.  Then, we can evaluate the integral by substituting u = tan x so that du = sec 2 x dx. Example 2

11. TANGENT & SECANT INTEGRALS We have: Example 2

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

13. TANGENT & SECANT INTEGRALS If we substitute u = sec x + tan x, then du = (sec x tan x + sec 2 x).  The integral becomes: ∫ (1/u) du = l n |u| + C

14. TANGENT & SECANT INTEGRALS Thus, we have:

15. Find Put, z= sin x dz= cos x dx SPECIAL TRIGONOMETRIC INTEGRALS Example 4     c xx x c zz z dzzz z xdxx x I             5 sin 3 2 53 2 21 1 cossin1 cos 53 53 42 2 2 2 2 2 2 5

16. TRIGONOMETRIC INTEGRALS Find ∫ sin4x cos5x dx Use this formula- IntegralIdentity a∫ sin m x cos n x dx b∫ sin m x sin n x dx c∫ cos m x cos n x dx

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

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