calculus

Question 1

improper integral

Answer 1

the expression to be integrated is undefined within the limits of integration or at one or both of the limits

Question 2

examples of improper integrals

Answer 2

• a limit is infinity

- 1/x is undefined at x=0

Question 3

solving improper integrals with limits of infinity

Answer 3

1. replace infinity with a variable, a
2. evaluate the integral using a and the other limit
3. see what happens when a -> ∞
4. if the expression with a tends to 0 the value of the integral is the expression without a else the value can’t be found

Question 4

solving improper integrals when the integral is undefined at a particular value

Answer 4

1. split the integral at the undefined point
2. replace the undefined limit with a a variable, a
3. evaluate the integral(s) with a
4. see what happens when a-> to the undefined value

Question 5

derivative of arcsin x

Answer 5

1/ √(1-x^2)

Question 6

derivative of arccos x

Answer 6

• 1/ √(1-x^2)

Question 7

derivative of arctan x

Answer 7

1/ (1+x^2)

Question 8

integral of 1/ √(a^2-x^2)

Answer 8

arcsin x/a + c

Question 9

integral of 1/ (a^2+x^2)

Answer 9

1/a arctan x/a + c

Question 10

volume of a solid formed by rotation about the x-axis

Answer 10

V = ∫ πy^2 dx

Question 11

volume of a solid formed by rotation about the y-axis

Answer 11

V = ∫ πx^2 dy

Question 12

mean value of a function

Answer 12

1/b-a ∫ f(x) dx

where b and a are the limits

Question 13

integral of cosecxcotx

Answer 13

-cosecx + c

Question 14

integral of secxtanx

Answer 14

secx + c

Question 15

integral of cosec^2x

Answer 15

-cotx

Question 16

derivative of tanx

Answer 16

sec^2x

Question 17

when to substitue trigonometric functions

Answer 17

when there are fractional powers of (a+x^2)

Question 18

volumes of revolution with two lines

Answer 18

square before you subtract

Question 19

derivative of sinh x

Answer 19

cosh x

Question 20

derivative of cosh x

Answer 20

sinh x

Question 21

derivative of sin x

Answer 21

cos x

Question 22

derivative of cos x

Answer 22

• sin x

Question 23

derivative of arcsinh x

Answer 23

1/ √(1+x^2)

Question 24

derivative of arccosh x

Answer 24

1/√(x^2 -1)

Question 25

derivative of arctanh x

Answer 25

1/ (1-x^2)

Question 26

integral of a 1/ √(a^2+x^2)

Answer 26

arcsinh(x/a) + c

Question 27

integral of 1/√(x^2 -a^2)

Answer 27

arccosh(x/a) + c

Question 28

integrating polar co-ordinates

Answer 28

1/2r^2dθ