# HW6 - Section 6.1 1. F ( x ) is an anti derivative of f ( x...

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Unformatted text preview: Section 6.1 1. F ( x ) is an anti derivative of f ( x ) if F ( x ) = f ( x ). Taking the derivative we find F ( x ) = x 2 + 4 x- 1 = f ( x ) so that F ( x ) is an anti derivative of f ( x ) . 3. F ( x ) is an anti derivative of f ( x ) if F ( x ) = f ( x ). Taking the derivative we find F ( x ) = (1 / 2)(2 x 2- 1)- . 5 (4 x ) = 2 x 2 x 2- 1 = f ( x ) so that F ( x ) is an anti derivative of f ( x ) . 5. G ( x ) is an anti derivative of f ( x ) if F ( x ) = f ( x ). Taking the derivative we find G ( x ) = 2 = f ( x ) so that G ( x ) is an anti derivative of f ( x ) . The set of all anti derivatives of f ( x ) is found by adding an arbitrary constant to G ( x ) . Thus any anti derivative of f ( x ) is of the form 2 x + C . 7. G ( x ) is an anti derivative of f ( x ) if F ( x ) = f ( x ). Taking the derivative we find G ( x ) = x 2 = f ( x ) so that G ( x ) is an anti derivative of f ( x ) . The set of all anti derivatives of f ( x ) is found by adding an arbitrary constant to G ( x ) . Thus any anti derivative of f ( x ) is of the form x 2 + C . 9. R 6 dx = 6 x + C. (Rule 1) 11. R x 3 dx = 1 1+3 x 1+3 + C = 1 4 x 4 + C. (Rule 2) 13. R x- 4 dx = 1 1+(- 4) x 1+(- 4) + C =- 1 3 x- 3 + C. (Rule 2) 15. R x 2 / 3 dx = 1 1+(2 / 3) x 1+(2 / 3) + C = 3 5 x 5 / 3 + C. (Rule 2) 17. R x- 5 / 4 dx = 1 1+(- 5 / 4) x 1+(- 5 / 4) + C =- 4 x- 1 / 4 + C. (Rule 2) 19. R 2 x 2 dx = R 2 x- 2 dx = 2 R x- 2 dx = (2) 1 1+(- 2) x 1+(- 2) + C = (2)(- 1) x- 1 + C =- 2 x- 1 + C. (Rules 2 and 3) 21. R t dt = R t 1 / 2 dt = ( ) 1 1+(1 / 2) t 1+(1 / 2) + C = ( ) 2 3 t 3 / 2 + C = 2 3 t 3 / 2 + C. (Rules 2 and 3) 23. R (3- 2 x ) dx = R 3 dx- 2 R x dx = 3 x- (2) 1 1+1 x 1+1 = 3 x- x 2 + C. (Rules 1, 2, 3, and 4) 25. R ( x 2 + x + x- 3 ) dx = R x 2 dx + R x dx + R x- 3 dx = 1 1+2 x 1+2 + 1 1+1 x 1+1 + 1 1+(- 3) x 1+(- 3) + C = 1 3 x 3 + 1 2 x 2 +- 1 2 x- 2 + C. (Rules 2 and 4) 27. R 4 e x dx = 4 R e x dx = 4 e x + C. (Rules 3 and 5) 29. R 1 + x + e x dx = R 1 dx + R x dx + R e x dx = x + 1 1+1 x 1+1 + e x + C = x + 1 2 x 2 + e x + C. (Rules 1, 2, 4, and 5) 31. R 4 x 3- 2 x 2- 1 dx = 4 R x 3 dx- 2 R x- 2 dx- R 1 dx = (4) 1 1+3 x 1+3- (2) 1 1+(- 2) x 1+(- 2)- x + C = x 4 + 2 x- 1- x + C. (Rules 1, 2, 3, and 4) 33. R x 5 / 2 + 2 x 3 / 2- x dx = R x 5 / 2 dx + 2 R x 3 / 2 dx- R x dx = 1 1+(5 / 2) x 1+(5 / 2) +(2) 1 1+(3 / 2) x 1+(3 / 2)- 1 1+1 x 1+1 + C = (2 / 7) x 7 / 2 +(4 / 5) x 5 / 2- (1 / 2) x 2 + C. (Rules 2, 3, and 4) 35. R x + 3 x dx = R x 1 / 2 dx + 3 R x- 1 / 2 dx = 1 1+(1 / 2) x 1+(1 / 2) + (3) 1 1+(- 1 / 2) x 1+(- 1 / 2) + C = (2 / 3) x 3 / 2 + 6 x 1 / 2 + C. (Rules 2, 3, and 4) 37. R u 3 +2 u 2- u 3 u du = R u 3 3 u + 2 u 2 3 u- u 3 u du = R u 2 3 + 2 u 3- 1 3 du = (1 / 3) R u 2 du +(2 / 3) R u du- R 1 3 du = (1 / 3) 1 1+2 u 1+2 + (2 / 3) 1 1+1 u 1+1- (1 / 3) u + C = (1 / 9) u 3 + (1 / 3) u 2- (1 / 3) u + C....
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## This note was uploaded on 02/14/2012 for the course MAC 2233 taught by Professor Smith during the Spring '08 term at University of Florida.

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HW6 - Section 6.1 1. F ( x ) is an anti derivative of f ( x...

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