Trig Integrals

Question 1

Integrate sin^4(x) dx.

Answer 1

(3/8)x - (1/4)sin(2x) + (1/32)sin(4x) + C

Question 2

Integrate sec^3(x) dx.

Answer 2

(1/2)(sec x tan x + ln|sec x + tan x|) + C

Question 3

Integrate tan^5(x) dx.

Answer 3

(1/4)tan^4(x) - (1/2)tan^2(x) - ln|cos x| + C

Question 4

Integrate sin^3(x) * cos^2(x) dx.

Answer 4

(1/5)cos^5(x) - (1/3)cos^3(x) + C

Question 5

Integrate cos(4x)cos(2x) dx.

Answer 5

(1/12)sin(6x) + (1/4)sin(2x) + C

Question 6

Integrate tan^3(x)sec^4(x) dx.

Answer 6

(1/6)tan^6(x) + (1/4)tan^4(x) + C

Question 7

Integrate sin^2(x)cos^4(x) dx.

Answer 7

(1/16)x - (1/64)sin(4x) + (1/48)sin^3(2x) + C (Use identities)

Question 8

Integrate csc^4(x) dx.

Answer 8

-cot(x) - (1/3)cot^3(x) + C

Question 9

Integrate sin(5x)sin(3x) dx.

Answer 9

(1/4)sin(2x) - (1/16)sin(8x) + C

Question 10

Integrate sec^6(x) dx.

Answer 10

tan(x) + (2/3)tan^3(x) + (1/5)tan^5(x) + C

Question 11

Integrate cos^5(x) dx.

Answer 11

sin(x) - (2/3)sin^3(x) + (1/5)sin^5(x) + C

Question 12

Integrate tan(x)sec^2(x) dx.

Answer 12

(1/2)tan^2(x) + C

Question 13

Integrate sin^2(x) dx.

Answer 13

(1/2)x - (1/4)sin(2x) + C

Question 14

Integrate cot^3(x) dx.

Answer 14

-(1/2)cot^2(x) - ln|sin x| + C

Question 15

Integrate sin(x)cos(3x) dx.

Answer 15

(1/4)cos(2x) - (1/8)cos(4x) + C

Question 16

Integrate sec^4(x)tan^2(x) dx.

Answer 16

(1/3)tan^3(x) + (1/5)tan^5(x) + C

Question 17

Integrate cos^4(x) dx.

Answer 17

(3/8)x + (1/4)sin(2x) + (1/32)sin(4x) + C

Question 18

Integrate sin^4(x)cos^2(x) dx.

Answer 18

(1/16)x - (1/64)sin(4x) - (1/48)sin^3(2x) + C

Question 19

Integrate csc^3(x) dx.

Answer 19

-(1/2)(csc x cot x + ln|csc x + cot x|) + C

Question 20

Integrate tan^2(x) dx.

Answer 20

tan(x) - x + C

Question 21

Integrate sin^2(x)cos^2(x) dx.

Answer 21

(1/8)x - (1/32)sin(4x) + C

Question 22

Integrate cos^2(3x) dx.

Answer 22

(1/2)x + (1/12)sin(6x) + C

Question 23

Integrate sec(x)tan^3(x) dx.

Answer 23

(1/3)sec^3(x) - sec(x) + C

Question 24

Integrate sin(2x)sin(x) dx.

Answer 24

(2/3)sin^3(x) + C

Question 25

Integrate cos^3(x) dx.

Answer 25

sin(x) - (1/3)sin^3(x) + C

Question 26

Integrate tan^4(x) dx.

Answer 26

(1/3)tan^3(x) - tan(x) + x + C

Question 27

Integrate sec^2(x)tan^3(x) dx.

Answer 27

(1/4)tan^4(x) + C

Question 28

Integrate sin(mx)cos(nx) dx (m != n).

Answer 28

-(cos((m+n)x)/(2(m+n))) - (cos((m-n)x)/(2(m-n))) + C

Question 29

Integrate csc^2(x)cot^2(x) dx.

Answer 29

-(1/3)cot^3(x) + C

Question 30

Integrate cos^6(x) dx.

Answer 30

(1/16)(5x + 8sin(2x) + (3/4)sin(4x) + (1/6)sin^3(2x)) + C