5.4 - d dx x 2 1 C = d dx x 2 1 1/2 C = 1 2 x 2 1 Â 1/2 Â¢...

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Unformatted text preview: d dx x 2 +1 + C = d dx x 2 +1 ( ) 1/2 + C = 1 2 x 2 +1 ( ) Â¡ 1/2 Â¢ 2 x = x x 2 +1 d dx x sin x +cos x + C = x cos x + sin x ( ) Â¢ 1 Â¡ sin x = x cos x d dx x a 2 a 2 Â¡ x 2 + C = 1 a 2 a 2 Â¡ x 2 Â¡ x ( Â¡ x ) a 2 Â¡ x 2 / a 2 Â¡ x 2 = 1 a 2 a 2 Â¡ x 2 ( ) + x 2 a 2 Â¡ x 2 ( ) 3/2 = 1 a 2 Â¡ x 2 ( ) 3 d dx Â¡ x 2 + a 2 a 2 x + C = Â¡ 1 a 2 d dx x 2 + a 2 x = Â¡ x x / x 2 + a 2 ( ) Â¡ x 2 + a 2 Â¢ 1 a 2 x 2 = Â¡ x 2 Â¡ x 2 + a 2 ( ) a 2 x 2 x 2 + a 2 = 1 x 2 x 2 + a 2 Â£ x Â¡ 3/4 dx = x Â¡ 3/4+1 Â¡ 3/4+1 + C = x 1/4 1/4 + C =4 x 1/4 + C Â£ 3 x dx = Â£ x 1/3 dx = x 4/3 4/3 + C = 3 4 x 4/3 + C Â£ x 3 +6 x +1 ( ) dx = x 4 4 +6 x 2 2 + x + C = 1 4 x 4 +3 x 2 + x + C Â£ x 1+2 x 4 ( ) dx = Â£ x +2 x 5 ( ) dx = x 2 2 +2 x 6 6 + C = 1 2 x 2 + 1 3 x 6 + C Â£ (1 Â¡ t ) 2+ t 2 ( ) dt = Â£ 2 Â¡ 2 t + t 2 Â¡ t 3 ( ) dt =2 t Â¡ 2 t 2 2 + t 3 3 Â¡ t 4 4 + C =2 t Â¡ t 2 + 1 3 t 3 Â¡ 1 4 t 4 + C Â£ x 2 +1+ 1 x 2 +1 dx = x 3 3 + x +tan Â¡ 1 x + C 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 1 Stewart Calculus ET 5e 0534393217;5. Integrals; 5.4 Indefinite Integrals and the Net Change Theorem Â¡ 2 Â¢ x ( ) 2 dx = Â¡ 4 Â¢ 4 x + x ( ) dx =4 x Â¢ 4 x 3/2 3/2 + x 2 2 + C =4 x Â¢ 8 3 x 3/2 + 1 2 x 2 + C Â¡ 3 e u + sec 2 u ( ) du = 3 e u + tan u + C Â¡ sin x 1 Â¢ sin 2 x dx = Â¡ sin x cos 2 x dx = Â¡ 1 cos x Â£ sin x cos x dx = Â¡ sec x tan xdx =sec x + C Â¡ sin 2 x sin x dx = Â¡ 2sin x cos x sin x dx = Â¡ 2cos xdx =2sin x + C Â¡ x x dx = Â¡ x 3/2 dx = 2 5 x 5/2 + C C =5 3 Â¢ 2 Â¢ 4 Â¡ (cos x Â¢ 2sin x ) dx =sin x +2cos x + C C =5 3 Â¢ 2 Â¢ 4 Â¡ 2 (6 x 2 Â¢ 4 x +5) dx = 6 Â£ 1 3 x 3 Â¢ 4 Â£ 1 2 x 2 +5 x 2 = 2 x 3 Â¢ 2 x 2 +5 x 2 =(16 Â¢ 8+10) Â¢ 0=18 Â¡ 3 1 (1+2 x Â¢ 4 x 3 ) dx = x +2 Â£ 1 2 x 2 Â¢ 4 Â£ 1 4 x 4 3 1 = x + x 2 Â¢ x 4 3 1 =(3+9 Â¢ 81) Â¢ (1+1 Â¢ 1)= Â¢ 69 Â¢ 1= Â¢ 70 Â¡ Â¢ 1 2 x Â¢ e x ( ) dx = x 2 Â¢ e x Â¢ 1 = 0 Â¢ 1 ( ) Â¢ 1 Â¢ e Â¢ 1 ( ) = Â¢ 2+1/ e 11. 12. 13. 14. 15. . The members of the family in the figure correspond to , , , , and . 16. . The members of the family in the figure correspond to , , , , and . 17. 18. 19. 2 Stewart Calculus ET 5e 0534393217;5. Integrals; 5.4 Indefinite Integrals and the Net Change Theorem Â¡ Â¢ 2 ( u 5 Â¢ u 3 + u 2 ) du = 1 6 u 6 Â¢ 1 4 u 4 + 1 3 u 3 Â¢ 2 =0 Â¢ 32 3 Â¢ 4 Â¢ 8 3 = Â¢ 4 Â¡ 2 Â¢ 2 (3 u +1) 2 du = Â¡ 2 Â¢ 2 9 u 2 +6 u +1 ( ) du = 9 Â£ 1 3 u 3 +6 Â£ 1 2 u 2 + u 2 Â¢ 2 = 3 u 3 +3 u 2 + u 2 Â¢ 2 =(24+12+2) Â¢ ( Â¢ 24+12 Â¢ 2)=38 Â¢ ( Â¢ 14)=52 Â¡ 4 (2 v +5)(3 v Â¢ 1) dv = Â¡ 4 (6 v 2 +13 v Â¢ 5) dv = 6 Â£ 1 3 v 3 +13 Â£ 1 2 v 2 Â¢ 5 v 4 = 2 v 3 + 13 2 v 2 Â¢ 5 v 4 =(128+104 Â¢ 20) Â¢ 0=212 Â¡ 4 1 t (1+ t ) dt = Â¡ 4 1 ( t 1/2 + t 3/2 ) dt = 2 3 t 3/2 + 2 5 t 5/2 4 1 = 16 3 + 64 5 Â¢ 2 3 + 2 5 = 14 3 + 62 5 = 256 15 Â¡ 9 2 t dt = Â¡ 9 2 t 1/2 dt = 2 Â£ 2 3 t 3/2 9 = 2 Â£ 2 3 Â£ 27 Â¢ 0=18 2 Â¡ Â¢ 1 Â¢...
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This note was uploaded on 12/08/2009 for the course MATH 101 taught by Professor Dr.tahir during the Fall '08 term at King Fahd University of Petroleum & Minerals.

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5.4 - d dx x 2 1 C = d dx x 2 1 1/2 C = 1 2 x 2 1 Â 1/2 Â¢...

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