Calculus Trigonometric functions

Question 1

f(x) = csc^-1(ax)

Answer 1

f^’(x) =

-1/ax² (1-(1/(ax)²))^1/2

Question 2

f(x) = sec^-1(ax)

Answer 2

f’(x) =

1/ax2 (1-(1/(ax)2))^1/2

Question 3

tan²(x)

Half Angle Formula

Answer 3

(1-cos(2x))/(1+cos(2x))

Question 4

cos²(x)

Half Angle

Answer 4

(1+cos(2x))/2

Question 5

sin²(x)

Half Angle

Answer 5

(1-cos(2x))/2

Question 6

sin(2x)

Answer 6

2sin(x)cos(x)

Question 7

cos(2x)

Answer 7

cos²(x) - sin²(x)

2cos²(x) - 1

1 - 2sin²(x)

Question 8

substitution for integral of:

dx/(a² + x²)dx

Answer 8

x = atan(@)

dx = asec²(@)d@

@ = arctan(x/a)

Question 9

substitution for integral of:

dx/(a²-x²)^1/2dx

Answer 9

x = asin(@)

dx = acos(@)d@

@ = arcsin(x/a)

Question 10

substitution for integral of:

(x²-a²)^1/2dx

Answer 10

x = asec(@)

dx = asec(@)tan(@)d@

@ = arcsec(x/a)

Question 11

derivative of:

tan(x)

Answer 11

sec²(x)

Question 12

derivative of:

cot(x)

Answer 12

-csc²(x)

Question 13

derivative of:

csc(x)

Answer 13

-csc(x)cot(x)

Question 14

derivative of:

sec(x)

Answer 14

sec(x)tan(x)

Question 15

derivative of:

cos^-1(ax)

Answer 15

• a/(1-(ax)²)^1/2

Question 16

derivative of:

sin^-1(ax)

Answer 16

a/(1-(ax)²))^1/2

Question 17

derivative of:

tan^-1(ax)

Answer 17

a/(1+(ax)²)

Question 18

integral of

cosx dx

Answer 18

sinx +c

Question 19

integeral of

sinx dx

Answer 19

-cosx

Question 20

integral of

sec² x dx

Answer 20

tanx + c

Question 21

integral of

csc²x dx

Answer 21

-cotx + c

Question 22

integral of

sec(x)tan(x) dx

Answer 22

sec(x) + c

Question 23

integral of

csc(x)cot(x) dx

Answer 23

-csc(x)

Question 24

integral of

tanx dx

Answer 24

ln|secx| + c

Question 25

integral of

cot(x) dx

Answer 25

ln|sinx| + c

Question 26

integral of

sec(x) dx

Answer 26

ln|secx + tanx| + c

Question 27

integral of

csc(x) dx

Answer 27

ln|csc(x) - cot(x)| + c