Derivatives and Integral Formulas

Question 1

d/dx(sinax)

Answer 1

acos(ax)

Question 2

d/dx(cosax)

Answer 2

-asin(ax)

Question 3

d/dx(tanax)

Answer 3

asec^2(ax)

Question 4

d/dx(cotx)

Answer 4

acsc^2(ax)

Question 5

d/dx(secax)

Answer 5

asecaxtanax

Question 6

d/dx(cscax)

Answer 6

-acscaxcotax

Question 7

d/dx(e^ax)

Answer 7

ae^ax

Question 8

d/dx(b^x)

Answer 8

b^xlnb

Question 9

d/dx(ln|x|)

Answer 9

1/x

Question 10

Power Rule: d/dx(x^n)

Answer 10

nx^n-1

Question 11

Product Rule: d/dx(f(x)*g(x))

Answer 11

f’(x)g(x)+g’(x)f(x)

Question 12

Quotient Rule: d/dx(f(x)/g(x))

Answer 12

f’(x)g(x)+g’(x)f(x)/(g(x))^2

Question 13

∫ cosaxdx

Answer 13

1/a(sinax)+C

Question 14

∫ sinaxdx

Answer 14

-1/a(cosax)+C

Question 15

∫ sec^(2)axdx

Answer 15

1/a(tanax)+C

Question 16

∫ csc^(2)axdx

Answer 16

-1/a(cotax)+C

Question 17

∫ secaxtanaxdx

Answer 17

1/a(secax)+C

Question 18

∫ cscaxcotaxdx

Answer 18

-1/a(cscax)+C

Question 19

∫ e^axdx

Answer 19

1/ae^ax+C

Question 20

∫ b^xdx

Answer 20

1/lnb(b^x)+C

Question 21

∫ (1/x)dx

Answer 21

ln|x|+C

Question 22

∫ (1/a^2+x^2)

Answer 22

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

Question 23

∫ (1/xsqrt(x^2-a^2))

Answer 23

1/a(sec^-1|x/a|+C

Question 24

∫ (1/sqrt(a^2-x^2))

Answer 24

sin-1(x/a)+C

Question 25

d/dx(sin^-1x)

Answer 25

1/sqrt(1-x^2)

Question 26

d/dx(cos^-1x)

Answer 26

-1/sqrt(1-x^2)

Question 27

d/dx(tan^-1x)

Answer 27

1/1+x^2

Question 28

d/dx(cot^-1x)

Answer 28

-1/1+x^2

Question 29

d/dx(sec^-1x)

Answer 29

1/xsqrt(x^2-1)

Question 30

d/dx(csc^-1x)

Answer 30

-1/xsqrt(x^2-1)

Question 31

rec sec(x)

Answer 31

1/(cosx)

Question 32

rec csc(x)

Answer 32

1/(sinx)

Question 33

rec cot(x)

Answer 33

1/(tanx)

Question 34

rec cos(x)

Answer 34

1/(secx)

Question 35

rec sin(x)

Answer 35

1/(cscx)

Question 36

rec tan(x)

Answer 36

1/(cotx)

Question 37

tan(x)

Answer 37

sinx/cosx

Question 38

cot(x)

Answer 38

cosx/sinx

Question 39

sin^2x+cos^2x

Answer 39

1

Question 40

1+tan^2x

Answer 40

sec^2x

Question 41

1+cot^2x

Answer 41

csc^2x

Question 42

sin^2(A)

Answer 42

1/2(1-cos(2A))

Question 43

cos^2(A)

Answer 43

1/2(1+cos(2A))

Question 44

sin(2A)

Answer 44

2sin(A)cos(A)

Question 45

cos(2A)

Answer 45

2cos^2(A)-1,1-2sin^2(A),cos^2(A)-sin^2(A)

Question 46

∫

Answer 46

integral sign

Question 47

∫f(x)dx what is f(x)

Answer 47

integrand

Question 48

∫tan(ax)dx

Answer 48

-1/aln|cos(ax)|+C

Question 49

∫cot(ax)dx

Answer 49

1/aln|sin(ax)|+C

Question 50

∫sec(ax)dx

Answer 50

1/aln|sec(ax)+tan(ax)|+C

Question 51

∫csc(ax)dx

Answer 51

1/aln|csc(ax)+cot(ax)|+C

Question 52

∫udv

Answer 52

uv-∫vdu

Question 53

acronym for integration by parts

Answer 53

LIATE, log, inverse trigonometric, algebraic, trignometric, exponential

Question 54

sin^m(x)cos^n(x) if power of sin is odd

Answer 54

save one sine factor with the dx, pythagorean identity, u-substitution

Question 55

sin^m(x)cos^n(x) if power of cosine odd

Answer 55

save one cosine factor with the dx, pythagorean identity, u-substitution

Question 56

sin^m(x)cos^n(x) all powers even

Answer 56

power reducing formula, simplify

Question 57

sec^m(x)tan^n(x) if the power of secant is even

Answer 57

save a factor of sec^2(x) with dx, Use sec^2(x)=1+tan^2(x), use u-substitution

Question 58

sec^m(x)tan^n(x) if the power of tangent is odd

Answer 58

save a factor of sec(x)tan(x) with dx, use tan^2(x)=sec^2(x)-1,use u-substitution

Question 59

∫tan^n(x)dx

Answer 59

(1/n-1)tan^n-1(x)-∫tan^n-2(x)dx

Question 60

∫sec^n(x)dx

Answer 60

(1/n-1)tan(x)sec^n-2(x)+(n-2/n-1)∫sec^n-2(x)dx

Question 61

to use trigonometric substitution which 3 expressions must be included

Answer 61

a^2-x^2,x^2-a^2,x^2+a^2

Question 62

a^2-x^2

Answer 62

x=asin(theta), dx=acos(theta)dx

Question 63

x^2-a^2

Answer 63

x=asec(theta), dx=asec(theta)tan(theta)dtheta

Question 64

a^2+x^2

Answer 64

x=atan(theta), dx=asec^2(theta)dtheta