Intergration by parts

These are questions on integration by parts. They normally focus on trigonometric, exponential functions and basic numbers

Transcript

1. Created by T. Madas INTEGRATION BY PARTS Created by T. Madas
2. Created by T. Madas Question 1 Carry out the following integrations: 1.  x e2 x dx = 2.  3 x cos 2 x dx = 3.  1 1 x sin 4 x dx = − x cos 4 x + sin 4 x + C 4 16 4.  −2 x sin 5 x dx = 5.  (1 − 2 x ) e− x dx = ( 2 x − 1) e− x + 2e− x + C 6.  1 2 2 x 2 e −3 x dx = − x 2 e −3 x − x e−3 x − e −3 x + C 3 9 27 7.  16 x3 ln x dx = 4 x 4 ln x − x 4 + C 8.  ln x dx = x ln x − x + C 9.  x cos 12 x dx = 2 x sin 12 x + 4cos 12 x + C 10.  ( 3x − 1) sin ( 3x − 1) dx = − ( 3x − 1) cos ( 3x − 1) + sin ( 3x − 1) + C ( ) 1 2x 1 2x xe − e + C 2 4 3 3 x sin 2 x + cos 2 x + C 2 4 2 2 x cos 5 x − sin 5 x + C 5 25 ( ) ( ) 1 3 Created by T. Madas 1 3
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4. Created by T. Madas Question 2 Carry out the following integrations: 1.  2 6 x e3 x dx = 2 x e3 x − e3 x + C 3 2.  4 12 x cos 3 x dx = 4 x sin 3 x + cos 3 x + C 3 3.  1 1 x sin 6 x dx = − x cos 6 x + sin 6 x + C 6 36 4.  − x sin 2 x dx = 5.  ( 2 − x ) e−3 x dx = − ( 2 − x ) e−3 x + e−3 x + C 6.  x 2 e4 x dx = 7.  x 2 e 2 dx = −2 x 2 e 2 − 8 x e 2 − 16e 2 + C 8.  25 x 4 ln x dx = 5 x5 ln x − x 5 + C 9.  24 x cos 23 x dx = 36 x sin 23 x + 54cos 23 x + C 10.  x 2 sin (1 − x ) dx = x 2 cos (1 − x ) + 2 x sin (1 − x ) − 2cos (1 − x ) + C 1 1 x cos 2 x − sin 2 x + C 2 4 1 3 1 9 1 2 4x 1 4x 1 4x x e − xe + e + C 4 8 32 −1 x −1 x ( ) −1 x ( ) −1 x ( ) Created by T. Madas
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6. Created by T. Madas Question 3 Carry out the following integrations: 1.  1 x e 4 x dx = 1 x e 4 x − 1 e 4 x + C 2 8 32 2.  5 5 5 x sin 4 x dx = − x cos 4 x + sin 4 x + C 4 16 3.  ( 2 x + 1) cos 2 x dx = ( 2 x + 1) sin 2 x + 4.  3 3 −3 x cos 4 x dx = − x sin 4 x − cos 4 x + C 4 16 5.  1 1 1 x 2 e−2 x dx = − x 2 e −2 x − x e −2 x − e −2 x + C 2 2 4 6.  1 2 2 x 2 sin 5 x dx = − x 2 cos5 x + x sin 5 x + cos5 x + C 5 25 125 7.  x 2 cos 13 x dx = 3 x 2 sin 13 x + 18 x cos 13 x − 54sin 13 x + C 8.  1 x 3 ln x dx = 1 x 4 ln x − 1 x 4 + C 2 8 32 9.  x ln 3 x dx = 10.  ln x ln x 1 dx = − 2 − 2 + C 3 x 2x 4x 1 2 1 cos 2 x + C 2 1 2 1 x ln 3 x − x 2 + C 2 4 Created by T. Madas
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8. Created by T. Madas Question 4 Carry out the following integrations: 1.  x e5 x dx = 2.  2 x cos 3 x dx = 3.  1 1 x sin 3 x dx = − x cos 3 x + sin 3 x + C 3 9 4.  x sin 4 x dx = 5.  2ln x dx = 2 x ln x − 2 x + C 6.  1 1 x 2 ln x dx = x 3 ln x − x3 + C 3 9 7.  x sin 12 x dx = 4sin 12 x − 2 x cos 12 x + C 8.  1 1 x sin ( 2 x − 1) dx = − x cos ( 2 x − 1) + sin ( 2 x − 1) + C 2 4 9.  ln x ln x 1 dx = − − +C 2 x x x 10.  x sec 2 x dx = x tan x − ln sec x + C ( ) 1 5x 1 5x xe − e +C 5 25 2 2 x sin 3 x + cos 3 x + C 3 9 1 1 sin 4 x − x cos 4 x + C 16 4 ( ) ( ) Created by T. Madas
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10. Created by T. Madas Question 5 Carry out the following integrations: 1.  x 2 sin x dx = − x 2 cos x + 2 x sin x + 2 cos x + C 2.  x3 ln x dx = 3.  sin x ln ( sec x ) dx = − cos x (1 + ln sec x ) + C 4.  x cos 5 x dx = 5.  1 2 2 x 2 sin 3 x dx = − x 2 cos 3 x + x sin 3 x + cos 3 x + C 3 9 27 6.  4 x e 3 dx = −3 ( 2 x + 3) e 3 + C 7.  1  1  1  1  x 2 cos  x  dx = 3 x 2 sin  x  + 18 x cos  x  − 54sin  x  + C 3  3  3  3  8.  2 x 2 sec2 x tan x dx = x 2 sec2 x − 2 x tan x + 2ln sec x + C 9.  x 2 e 2 dx = ( 2 x 2 − 8 x + 16 ) e 2 + C 10.  x sec x tan x dx = x sec x − ln sec x + tan x + C −2x 1x 1 4 1 x ln x − x 4 + C 4 16 1 1 x sin 5 x + cos 5 x + C 5 25 −2x 1x Created by T. Madas
11. Created by T. Madas Question 6 Carry out the following integrations: 1.  x 2 e 4 dx = −4 e 4 ( x 2 + 8 x + 32 ) + C 2.  x 2 e− x dx = − e− x ( x 2 + 2 x + 2 ) + C 3.  e x cos x dx = 4.  ( ln x ) dx = x ( ln x ) − 2 x ln x + 2 x + C 5.  1 e x sin x dx = e x ( sin x − cos x ) + C 2 6. 1 x + 5 x − 2 ) e dx = e ( 4 x + 14 x − 14 x − 1) + C (  8 7.  1 1 1 x cos 2 x dx = x 2 + x sin 2 x + cos 2 x + C 8 4 8 8.  x ln 2 x 3 dx = −1 x −1 x 1 x e ( cos x + sin x ) + C 2 2 3 2 2 2x 2x 3 2 3 2 x ln 2 ( 2ln x − 1) + C 4 Created by T. Madas
12. Created by T. Madas Question 7 Carry out the following integrations, to the answer given: 1. ln 2  x e 2 x dx = ln 4 − 0 2. π  3 6 x sin 3 x dx = 0 3.  2 x 2 cos x dx =  e  1 1 2 ( e + 1) 4 4 x e3 x dx = 4 ( 2e3 + 1) 9 0 6. π  4 x sin 4 x dx = 0 7. 1 2 (π − 8 ) 4 x ln x dx = 1 5. 2π 3 π 0 4.  2  1 π 16 x 3 ln x dx = 4 ln 2 − 1 8. x e −2 x dx = 0 9.   15 16 1 1 − 3e −2 ) ( 4 π 4 0 10. 3 4 12 x cos 2 x dx = 3 (π − 2 ) 2 π 2 4 x sin 2 x dx = π − 1 π 4 Created by T. Madas
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14. Created by T. Madas Question 8 Carry out the following integrations, to the answer given: 1. π  3 x sin 3 x dx = 0 2. π 9 π  4 2 x cos 4 x dx = − 0 3.  ln 2  e 1 4 4 x e− x dx = 2 − ln 4 0 4. ln x dx = 1 1 5.  π 2 x sin 2 x dx = 0 6.  ln 4  π π 4 1x x e 2 dx = 8ln 2 − 4 0 7. 1  x cos  x  dx = 2 2 (π + 4 ) − 16 4  0 1 8. ( 2 x + 1) e 2 x dx = e 2 0 9. 10.  1  0 1 3  x ln x dx =  2 − 1 4e 1  e 3ln ( 2 x + 3) dx = 32 ( ln 27 − 2 ) REQUIRES ADDITIONAL TECHNIQUES −1 Created by T. Madas
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16. Created by T. Madas Question 9 Carry out the following integrations, to the answer given: 1.  π 4 x sec2 x dx = 0 2. 3.  1 (π − ln 4 ) 4 2 ln x 1 2 dx = ( ln 2 ) x 2 1  π 0 2 x sin 2 x dx = 1 2 (π + 4 ) 16 Created by T. Madas