Integrals: Basic Forms

Question 2

∫ u dv

Answer 2

uv - ∫ v du

Question 3

∫ u^n du

Answer 3

(u^(n+1))/(n+1), n != 1

ln u, n == 1

Question 4

∫ du/u

Answer 4

ln |u|

Question 5

∫ e^u du

Answer 5

e^u

Question 6

∫ a^u du

Answer 6

(a^u)/(ln a)

Question 7

∫ sin u du

Answer 7

-cos u

Question 8

∫ cos u du

Answer 8

sin u

Question 9

∫ sec^2 u du

Answer 9

tan u

Question 10

∫ csc^2 u du

Answer 10

-cot u

Question 11

∫ (sec u)(tan u) du

Answer 11

sec u

Question 12

∫ (csc u)(cot u) du

Answer 12

-csc u

Question 13

∫ tan u du

Answer 13

ln |sec u|

Question 14

∫ cot u du

Answer 14

ln |sin u|

Question 15

∫ sec u du

Answer 15

ln |sec u + tan u|

Question 16

∫ csc u du, uεR

Answer 16

ln |csc u – cot u|

Likes most if not all of these antiderivatives, this answer holds iff uεR. If u has an imaginary component, the antiderivative is:
ln(sin(u/2)) - ln(cos(u/2))

Question 17

∫ du/√(a^2 – u^2)

Answer 17

sin^-1 u/a , a > 0

Question 18

∫ du/(a^2 + u^2)

Answer 18

(1/a)(tan^-1 u/a)

Question 19

du
∫ —————
(u)√(u^2 – a^2)

Answer 19

(1/a)(sec^-1 u/a)

Question 20

∫ du/(a^2 – u^2)

Answer 20

(1/2a)(ln |(u+a)/(u-a)|)

Question 21

∫ du/(u^2 – a^2)

Answer 21

(ln |(u-a)/(u+a)|)/2a

Question 22

∫ (v^u)(ln v) dv

Answer 22

u^v

Question 23

∫ csc du, uεC

Answer 23

ln(sin(u/2)) - ln(cos(u/2))

Note that with complex numbers, log(xy) != log(x)+log(y)

Question 24

Lol?

Answer 24

Ye