Integration by parts trigonometric functions

These are questions on integration by parts. They normally focus on trigonometric functions .

Transcript

1. Created by T. Madas INTEGRATION BY TRIGONOMETRIC IDENTITIES Created by T. Madas
2. Created by T. Madas Question 1 Carry out the following integrations: 1. ∫ 3sin 2 x dx = 2. ∫ 4cos 2 x dx = 2 x + sin 2 x + C 3. ∫ 3 3sin x cos x dx = − cos 2 x + C 4 4. ∫ ( 2 − 3sin x ) dx = 5. ∫ (1 − cos 2 x ) dx = 6. ∫ 2 tan 2 x dx = 2 tan x − 2 x + C 7. ∫ 5cot 2 x dx = −5cot x − 5 x + C 8. ∫ ( 2 tan x − cot x ) dx = 4 tan x − cot x − 9 x + C 9. ∫ 4sin x dx = 4sec x + C cos 2 x 10. ∫ cos x 1 dx = − cosec x + C 2 3sin x 3 3 3 x − sin 2 x + C 2 4 2 2 17 9 x + 12cos x − sin 2 x + C 2 4 3 1 x − sin 2 x + sin 4 x + C 2 8 2 Created by T. Madas
3. Created by T. Madas Created by T. Madas
4. Created by T. Madas Question 2 Carry out the following integrations: 1. ∫ ( 2 + sin x ) dx = 2. ∫ sin x (1 + sec 2 x ) dx = sec x − cos x + C 3. ∫ (1 − 2 cos x ) dx = 3x − 4sin x + sin 2 x + C 4. ∫ 1 dx = − cot x + C cos x tan 2 x 5. ∫ 2 + 2 tan 2 x dx = 2 tan x + C 6. ∫ 1 + cos x dx = − cot x − cosec x + C sin 2 x 7. ∫ (1 + cos x ) dx = −2 cot x − x − 2 cosec x + C 8. ∫ 4 cos 2 x dx = 2 x + sin 2 x + C 9. ∫ 3cot 2 x dx = −3cot x − 3 x + C 10. ∫ ( 2 cos x − 3sin x ) dx = 2 9 1 x − 4 cos x − sin 2 x + C 2 4 2 2 2 sin 2 x 2 13 5 x − sin 2 x + 3cos 2 x + C 2 4 Created by T. Madas
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6. Created by T. Madas Question 3 Carry out the following integrations: 1. ∫ sin 2 x cosec x dx = 2 sin x + C 2. ∫ 1 + sin x dx = sec x + tan x + C cos 2 x 3. ∫ tan 2 x dx = tan x − x + C 4. ∫ (1 + sin x ) dx = 2 tan x + 2sec x − x + C 5. ∫ cos 2 x dx = x + cos x + C 1 + sin x 6. ∫ 1 dx = cosec x − cot x + C 1 + cos x 7. ∫ (1 + 2 cos x ) dx = − 5 cot x − 4 cosec x − 4 x + C 8. ∫ 1 1 sin x sin 3 x dx = sin 2 x − sin 4 x + C 4 8 9. ∫ sin 2 2 x dx = 10. ∫ 2 cos 3 x sin x dx = 2 cos 2 x 2 3sin 2 x 3 3 3 1 1 x − sin 4 x + C 2 8 1 1 cos 2 x − cos 4 x + C 2 4 Created by T. Madas
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8. Created by T. Madas Question 4 Carry out the following integrations: 1. ∫ cos 2 x 1 dx = − cosec 2 x + C 2 1 − cos 2 x 2 2. ∫ 1 cot 2 3 x dx = − x − cot 3 x + C 3 3. ∫ sin 2 x sec x dx = −2 cos x + C 4. ∫ 1  x dx = ln tan   + sec x + C 2 sin x cos x 2 5. ∫ 1 dx = − x − cot x − cosec x + C sec x − 1 6. ∫− 7. ∫ ( 2 cos x − 3) dx = 11x + sin 2 x − 12sin x + C 8. ∫ ( 3sin x − cos x ) dx = 5 x − 2sin 2 x + 9. ∫ 1 dx = ln sec x + tan x − cosec x + C cos x sin 2 x 10. ∫ sin 2 x sec2 x dx = tan x − x + C 1 cot 2 x dx = 2 x + cot x + C 2 2 3 cos 2 x + C = 5 x − 2sin 2 x − 3sin 2 x + C 2 Created by T. Madas
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10. Created by T. Madas Question 5 Carry out the following integrations: 1. ∫ 1 1 sin 3 x cos 2 x dx = − cos x − cos5 x + C 2 10 2. ∫ 1 1 dx = − ln cosec 2 x + cot 2 x + C = ln tan x + C sin x cos x 2 3. ∫ 1 dx = sec x + tan x + C 1 − sin x 4. ∫ sin 2 2 x dx = 5. ∫ cos 2 x dx = 2 x − tan x + C cos 2 x 6. ∫ cos 2 x sin 2 x dx = 7. ∫ ( sin x + 2 cos x ) dx = 8. ∫ 1 dx = −2 cot 2 x + C sin x cos 2 x 9. ∫ 10. ∫ 1 1 x − sin 4 x + C 2 8 2 1 1 x − sin 4 x + C 8 32 5 3 x + 2 sin 2 x + sin 2 x + C 2 4 2 2 x sin 2 x + ( cos x − 1) dx = −4 cos   + C 2 1 − cos x  x dx = 2 tan   − x + C = −2 cot x − x + 2 cosec x + C 1 + cos x 2 Created by T. Madas
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12. Created by T. Madas Question 6 Carry out the following integrations: 1. ∫ 1 + sin x dx = 2 tan x − x + 2sec x + C 1 − sin x 2. Created by T. Madas
13. Created by T. Madas Question 7 Carry out the following integrations: 1. ∫ π 2 4sin 2 x dx = π 0 2. ∫ π 6 24cos 2 x dx = π + 3 0 3. ∫ π 6 8sin x cos x dx = 1 0 4. ∫ π 2 2 (1 − sin x ) dx = 0 5. ∫ 3π −4 2 π 6 (1 − cos3x ) 2 dx = π 0 6. ∫ 4 − 2 3 π 4 4 tan 2 x dx = 4 − π 0 7. ∫ π 3 2 ( 3cot x + tan x ) dx = π 2 10 3 − π 3 ( ) 6 8. ∫ π 4 2 ( sec x + 4cos x ) dx = 4π + 5 0 9. 10. ∫ π sin x dx = 1 2 0 cos x ∫ 3 π cos x dx = 2 − 1 2 π sin x 2 4 Created by T. Madas
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15. Created by T. Madas Question 8 Carry out the following integrations: 1. ∫ π 4 cos 2 x dx = 0 2. ∫ π 2 sin 2 x dx = 0 3. ∫ 1 (π + 2 ) 8 π 4 π 2 ( 2 sin x − 3cos x ) 2 dx = 0 4. 5π 3 ∫ (1 − 2 cos x ) π 2 1 (13π − 24 ) 4 dx = 4π + 3 3 3 5. ∫ π 4 tan 2 x dx = 0 6. ∫ 1 (4 −π ) 4 π 6 sin x sin 3 x dx = 0 7. ∫ 3 16 π 1 dx = 1 + 3 0 1 − sin x 3 π 8. ∫ (1+ tan 2x ) dx = 2 + ln 4 2 2 0 9. ∫ π 2 0 10. ∫ cos3 x dx = 2 3 π 6 π cot 2 2 x dx = 1 3 π − − 2 6 24 8 Created by T. Madas
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17. Created by T. Madas Question 9 Carry out the following integrations: 1. ∫ π 12 1 ( π − 3) 4 6sin 2 θ dθ = 0 2. ∫ π 6 sin 3 θ dθ = 0 3. ∫ 5 24 π 12 10sin 8θ cos 2θ dθ = 0 4. ∫ ∫ ( ) π 4 2 5 8 ( cos x + sec x ) dx = (π + 2 ) 0 5. 1 16 + 3 3 12 π 2 1 8 ( sin x + cot x ) dx = ( 26 − π − 4 2 ) π 2 4 Created by T. Madas