MA1014_HW4r_201012

MA1014_HW4r_201012 - Mathematics for Engineering II...

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Unformatted text preview: Mathematics for Engineering II Homework 4: Integration by substitution+ From the following integrals, solve only excersices: 1, 7, 11, 13, and 21. Use the substitution plus technique (u+). x 1. 2. € ∫ ( x + 1) x 2 dx dx ln x + 1 + 1 +c x +1 ∫ (1 − x ) ∫ ∫ 2 3. € x dx (2 x + 1) 5 x dx (1 − 2 x ) 5 € € € 1 + ln1 − x + c 1− x − 1 1 + +c 3 12(2 x + 1) 16(2 x + 1) 4 4. € 1 1 − +c 4 16(1 − 2 x ) 12(1 − 2 x ) 3 5. € ∫ (7 x + 4) ∫ (2 − 7 x ) ∫x ∫x ∫x x2 x 2 10 dx € Ⱥ 1 Ⱥ 1 1 16 + − +c Ⱥ− 7 8 9 Ⱥ 343 Ⱥ 7( 7 x − 4 ) ( 7 x + 4 ) 9( 7 x + 4 ) Ⱥ − Ⱥ 1 Ⱥ 2 4 + + ln 2 − 7 x Ⱥ + c Ⱥ− 2 343 Ⱥ (2 − 7 x ) 2 − 7 x Ⱥ 5 3 6. 3 dx € € 7. x + 1dx € 8. € 1 − x dx € € 2 2 ( x + 1) 2 − ( x + 1) 2 + c 5 3 2 2 − (1 − x ) 2 + (1 − x ) 2 + c 3 5 5 3 Ⱥ 1 Ⱥ2 4 Ⱥ ( 3 x + 2) 2 − ( 3 x + 2) 2 Ⱥ + c 9 Ⱥ5 3 Ⱥ 3 5 Ⱥ 1 Ⱥ 1 2 Ⱥ( 3 − 4 t ) − ( 3 − 4 t ) 2 Ⱥ + c 4 Ⱥ 5 Ⱥ 3 5 9. € 3 x + 2 dx € 10. € ∫ 2t 3 − 4 t dt € 11. € ∫ ( x + 1) ∫ (2 x + 1) ∫ (x 2 1 − x dx € 4 2 − (1 − x ) 2 + (1 − x ) 2 + c 3 5 4 34 ( x + 9) 2 − ( x + 9) 2 + c 5 3 7 5 3 Ⱥ 1 Ⱥ 2 4 16 2 2 Ⱥ ( 3 x + 1) − ( 3 x + 1) − ( 3 x + 1) 2 Ⱥ + c 27 Ⱥ 7 5 3 Ⱥ 5 3 3 5 12. € x + 9 dx € 13. € − 1) 3 x + 1dx € € € fpm@itesm.mx Mathematics for Engineering II Homework 4: Integration by substitution+ 14. ∫ (x ∫ ∫ ∫ ∫ ∫ ∫ 2 + 1) 2 x + 1dx 1 ( 2 x + 1) − ( 2 x + 1) + ( 2 x + 1) 2 + c 72 52 2 1 7 2 2 5 2 3 € 15. x dx x −1 x dx 1− x 2x + 1 dx 2x −1 5x + 1 dx 6 x − 20 x2 x − 11 x2 2 x − 21 dx dx € 2 ( x − 1) 2 + 2( x − 1) 2 + c 3 2 −2(1 − x ) + (1 − x ) 2 + c 3 1 (2 x − 1) 2 + 2(2 x − 1) 2 + c 3 5 53 (6 x − 20) 2 + (6 x − 20) 2 + c 54 9 2 4 ( x + 11) 2 + ( x + 11) 2 + 22( x + 11) 2 + c 5 3 1 21 21 (2 x − 21) 2 + (2 x − 21) 2 + (2 x − 21) 2 + c 20 6 4 32 3 ( x + 1) 3 − ( x 2 + 1) 3 + c 10 4 52 25 125 2 ( r + 5) 5 − ( r 2 + 5) 5 + ( r + 5) 5 + c 28 9 8 14 9 4 5 2 5 3 1 5 3 1 3 1 3 1 1 2 3 3 1 16. € € 17. € € 18. € € 19. € € 20. € € 21. € ∫ ∫ x3 3 x2 +1 r5 r2 + 5 w dx € 22. 5 dx € € 23. € 24. € ∫ w + 1 dw ∫ α +9 dα α+9 dµ µ−3 µ 2 w − 2 arctan w + c € € 2 α + 9 ln α + 9 − 6 arctan 1 1 1 1 α +c 3 25. € ∫ 2µ 2 + 3µ 3 + 6µ 6 + 6 ln µ 6 − 1 + c € € € fpm@itesm.mx ...
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