Differentiation And Integration Rules

Question 1

∫x^n dx = ?
(x^n)’ = ?

Answer 1

∫x^n dx = (1/n+1)x^n+1 + C
(x^n)’ = nx^n-1

Question 2

∫(1/x)dx = ?
(ln|x|)’ = ?
(lnx)’ = ?

Answer 2

∫(1/x)dx = ln|x| + C
(ln|x|)’ = 1/x
(lnx)’ = 1/x

Question 3

∫(e^x)dx = ?
(e^x) = ?

Answer 3

∫(e^x)dx = e^x + C
(e^x) = e^x

Question 4

∫cosxdx = ?
(cosx)’ = ?

Answer 4

∫cosxdx = sinx + C
(cosx)’ = -sinx

Question 5

∫sinxdx = ?
(sinx)’ = ?

Answer 5

∫sinxdx = -cosx + C
(sinx)’ = cosx

Question 6

∫sec^2 x dx = ?
(tanx)’ = ?

Answer 6

∫sec^2 x dx = tanx + C
(tanx)’ = sec^2 x

Question 7

∫secxtanxdx = ?
(secx)’ = ?

Answer 7

∫secxtanxdx = secx + C
(secx)’ = secxtanx

Question 8

∫tanxdx = ?

Answer 8

∫tanxdx = ln|secx| + C

Question 9

∫secxdx = ?

Answer 9

∫secxdx = ln|secx + tanx| + C

Question 10

∫csc^2 x dx = ?
(cotx)’ = ?

Answer 10

∫csc^2 x dx = -cotx + C
(cotx)’ = -csc^2 x

Question 11

∫cscxcotxdx = ?
(cscx)’ = ?

Answer 11

∫cscxcotxdx = -cscx + C
(cscx)’ = -cscxcotx

Question 12

∫cotxdx = ?

Answer 12

∫cotxdx = ln|sinx| + C

Question 13

∫cscxdx = ?

Answer 13

∫cscxdx = ln|cscx - cotx|+ C
= -ln|cscx + cotx| + C

Question 14

∫(1/sqrt(a^2 - x^2))dx = ?
(sin^-1 x)’ = ?

Answer 14

∫(1/sqrt(a^2 - x^2))dx = sin^-1 (x/a) + C
(sin^-1 x)’ = 1/sqrt(1 - x^2)

Question 15

∫(1/sqrt(a^2 + x^2))dx = ?
(tan^-1 x) = ?

Answer 15

∫(1/sqrt(a^2 + x^2))dx = (1/a)tan^-1 (x/a) + C
(tan^-1 x) = 1/(1 + x^2)