Integration Types for Leaving cert. Honours Mathematics


Thomas O'sullivan 1997

Integration Types

 

The following are 20 integration types essential for question 8 on both papers I and II of the Leaving Certificate Honours Mathematics Examination.

 

 

1. y =ò(axn + bxn-1 + cxn-2 +....)dx y = ò(2x3 + 4x2 + 6x)dx

 

2. y = ò(2x + 3)4dx (Let u = 2x + 3)

 

3. (Let u = x2 + 1)

 

4. y = ò(2x + 1)(2x - 4)3dx (Let u = 2x – 4 and find a value for 2x +1 in terms of u)

 

5. y = òsin mx dx or òcos mx dx or .... (let u = mx)

 

6. y = òsin 3x cos x dx .....(change to sum) note also òsin2xcos2x dx

 

7. y = òsin2x dx = ò˝ ( 1- cos 2x)dx or y = òcos2x dx

 

8. y = òsin2xcos3x dx... (Cos3x = cos x (1 - sin2x))

 

9. y = òenxdx (Let u = nx)

 

10. e.g. or y = ò tan x dx …….solution in the form loge f(x) + c

(let u = denominator)

11. Divide numerator by denominator

 

12. (Let u = logex )

also (Let )

 

13. () (Let u = 1+ex)

 

14. y = ò 2x dx let u = 2x then logeu = x loge2 differentiating gives

which implies that dx substitute ....

 

15. is a sin-1x problem

 

16. becomes (tan-1 function) (Let u = x –1)

 

17.

(Let u = x + 1) (sin-1 function)

 

18. (Let )

19. and (Let u = sin x) (tan-1 function)

 

 

 

Integration by parts is examinable on paper II in the option section:

 

 

20. y = òx2 logex dx; y =òxcos2x dx; y = òx2e4xdx; y = òx2sinx dx; y = òe2xsinx dx By parts.

 

Remember, the order of substitution is given by: L I A T E; Logs, indices, algebra, trigonometry and exponentials

 

© T O’Sullivan A Phabolous FIsics Production January 1994



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