useful formulae, derivatives, anti-derivatives, etc.

Question 1

What are the three half angle formulae?

Answer 1

sin(x/2) = √[(1-cosx)/2]
cos(x/2) = √[(1+cosx)/2]
tan(x/2) = (1-cosx)/sinx

Question 2

d/dx (sinx) = ?

Answer 2

cosx

Question 3

d/dx (cosx) = ?

Answer 3

-sinx

Question 4

d/dx (tanx) = ?

Answer 4

sec²x

Question 5

d/dx (secx) = ?

Answer 5

secxtanx

Question 6

d/dx (aˣ) = ?

Answer 6

aˣln(a)

Question 7

d/dx (logₐx) = ?

Answer 7

1/xln(a)

Question 8

∫cosx dx = ?

Answer 8

sinx + c

Question 9

∫sinx dx = ?

Answer 9

-cosx + c

Question 10

∫sec²x dx = ?

Answer 10

tanx + c

Question 11

∫secxtanx dx = ?

Answer 11

secx + c

Question 12

∫tanx dx = ?

Answer 12

ln|secx| + c

Question 13

∫aˣ dx = ?

Answer 13

(aˣ)/lna + c

Question 14

∫lnx dx = ?

Answer 14

xlnx - x + c

Question 15

∫xeˣ dx = ?

Answer 15

(x-1)eˣ + c

Question 16

∫1/xlnx dx = ?

Answer 16

ln|lnx| + c

Question 17

What would be the trig substitution for √(a²-b²x²) ?

Answer 17

x=(a/b)sinθ
and
cos²θ=1-sin²θ

Question 18

What would be the trig substitution for √(b²x²-a²) ?

Answer 18

x=(a/b)secθ
and
tan²θ=sec²θ-1

Question 19

What would be the trig substitution for √(a²+b²x²) ?

Answer 19

x=(a/b)tanθ
and
sec²θ=1+tan²θ