Antiderivatives

Question 1

Anti-derivative of cos x

Answer 1

sin x + C

Question 2

Anti-derivative of sin x

Answer 2

-cos x + C

Question 3

Anti-derivative of sec^2(x)

Answer 3

tan x + C

Question 4

Anti-derivative of csc^2(x)

Answer 4

-cot x + C

Question 5

Anti-derivative of (tan x)(sec x)

Answer 5

sec x + C

Question 6

Anti-derivative of (cot x)(csc x)

Answer 6

-csc x + C

Question 7

Anti-derivative of 1 / \sqrt{1 - x^2}

Answer 7

arc sin (x) + C

Question 8

Anti-derivative of - 1 / \sqrt{1 - x^2}

Answer 8

arc cos (x) + C

Question 9

Anti-derivative of 1 / (a^2 + x^2)

Answer 9

1/a * arc tan (x/a) + C

Question 10

Anti-derivative of - 1 / (a^2 + x^2)

Answer 10

1/a * arc cot (x/a) + C

Question 11

Anti-derivative of 1 / x^2 \sqrt{x^2 - a^2}

Answer 11

1/a arc sec (x/a) + C

Question 12

Anti-derivative of - 1 / x^2 \sqrt{x^2 - a^2}

Answer 12

1/a arc csc (x/a) + C

Question 13

Anti-derivative of x^n

Answer 13

x^{n+1} / (n+1) + C

Question 14

Anti-derivative of 1 / x

Answer 14

ln |x| + C

Question 15

Anti-derivative of 1 / (ax + b)

Answer 15

1/a ln |ax + b| + C

Question 16

Anti-derivative of tan x

Answer 16

• ln |cos x| + C

Question 17

Anti-derivative of cot x

Answer 17

ln |sin x| + C

Question 18

Anti-derivative of sec x

Answer 18

ln |sec x + tan x| + C

Question 19

Anti-derivative of csc x

Answer 19

• ln |csc x + cot x| + C

Question 20

Anti-derivative of sin(mx)*sin(nx)

Answer 20

• sin(mx+nx) / 2(m+n) + sin(mx-nx) / 2(m-n) + C

Question 21

Anti-derivative of cos(mx)*cos(nx)

Answer 21

sin(mx+nx) / 2(m+n) + sin(mx-nx) / 2(m-n) + C

Question 22

Anti-derivative of sin(mx)*cos(nx)

Answer 22

• cos(mx+nx) / 2(m+n) - cos(mx-nx) / 2(m-n) + C

Question 23

Anti-derivative of a^x

Answer 23

a^x / ln a + C

Question 24

Anti-derivative of ln x

Answer 24

x*ln x - x + C

Question 25

Anti-derivative of arc sin x

Answer 25

x*arc sin x + \sqrt{1-x^2} + C

Question 26

Anti-derivative of arc cos x

Answer 26

x*arc cos x - \sqrt{1-x^2} + C

Question 27

Anti-derivative of arc tan x

Answer 27

x*arc tan x + ln[\sqrt{1+x^2} + C

Question 28

Anti-derivative of arc cot x

Answer 28

x*arc cot x + ln[\sqrt{1+x^2} + C

Question 29

Anti-derivative of arc sec x

Answer 29

x*arc sec x - ln[x + \sqrt{x^2-1}] + C

Question 30

Anti-derivative of arc csc x

Answer 30

x*arc csc x + ln[x+\sqrt{x^2-1}] + C