calculus

Question 1

d/dx coth x

Answer 1

-cosech^2 x

Question 2

d/dx tanh

Answer 2

sech^2 x

Question 3

d/dx cosech x

Answer 3

-cosechx cothx

Question 4

d/dx arsinh x

Answer 4

1/sqrt(x^2 + 1)

Question 5

d/dx arcosh x

Answer 5

(x^2 -1)^-1/2

Question 6

d/dx artanh x

Answer 6

1/(1-x^2)

Question 7

∫ tanh x dx

Answer 7

ln coshx

Question 8

∫coth x dx

Answer 8

ln sinhx

Question 9

∫ f’(x)f(x)^n dx

Answer 9

f(x)^(n+1)/n+1

Question 10

osborne’s rule

Answer 10

a product of two sines change the sign

Question 11

volume of revolution 2π around x-axis and what is condition

Answer 11

π ∫ y^2 dx
only counts the +ve part of graph hence the square

Question 12

volume of revolution 2π about y axis

Answer 12

π ∫ x^2 dy
only +ve part of graph hence square

Question 13

1st order ODE form

Answer 13

dy/dx + p(x)y = q(x)

Question 14

1st order ODE process

Answer 14

multiply by I (intergrating factor)
I = e^∫ p(x) dx
then reverse product rule
then intergrate RHS

Question 15

2nd order ODE process

Answer 15

auxiliary eqn
complementary function
trial intergral for RHS
then general soln is the complementary function + trial intergral

Question 16

auxiliary equation has distinct roots a,b

Answer 16

complementary function : Ae^ax +Be^bx

Question 17

auxiliary equation has repeated root a

Answer 17

complementary function : e^ax (A +Bx)

Question 18

auxiliary equation has complex roots a ± bi

Answer 18

complementary function : e^ax ( Acos bθ + Bsin bθ)

Question 19

trial intergral for some polynomial of degree n

Answer 19

polynomial of degree n with coefficients a,b,c etc

Question 20

trial intergral for some function pe^kx

Answer 20

qe^kx

Question 21

trial intergral for some function Acos kθ + Bsin kθ

Answer 21

pcos kθ + qsin kθ