Calculus

Question 1

How do you know if an integral is an improper integral?

Answer 1

• If one or both of the limits is infinite
• If f(x) is undefined at any point in the interval or at the limits

Question 2

How do you know if an improper integral is convergent or divergent?

Answer 2

If the integral tends to infinity then it is divergent

If the integral has a finite value then it is convergent

Question 3

How do you deal with an integral where one of the limits is infinite?

Answer 3

• Write your integral as, lim t→∞ ∫f(x) swapping the infinity limit for a t
• Solve your integral as normal Note: make sure when you sub in your limits you simplify fully collecting all like terms
• State that as t→∞ then your t term either →∞ or →0
• Now state wether your integral is convergent or divergent, if it’s convergent state its final value

Question 4

How do you deal with an integral where one of the limits is undefined?

Answer 4

• State which limit (value of x) f(x) is undefined at
• Rewrite your integral as lim t→(undefined limit)^+or- ∫f(x) swapping the undefined limit for a t
• Solve your integral as normal Note: make sure when you sub in your limits you simplify fully collecting all like terms
• State that as t→∞ then your integral either →limit^+or- or →∞
• Now state wether your integral is convergent or divergent, if it’s convergent state its final value

Note if your upper limit is swapped for a t then the t→limit from the negative side or if your lower limit is swapped for a t then the t→limit from the positive side

Question 5

How do you deal with an integral that is not defined at a point between the limits? e.g an integral thats not defined at x=2 between 1 and 3

Answer 5

• State the value that the integral is undefined at i.e x=2
• Split the integral into two integrals i.e
• ∫₁² + ∫₂³
• Swap the undefined point for a t and introduce your limits
• Solve each integral as normal

Question 6

How do you deal with an integral where both limits are infinite? i.e
∫_-∞^∞

Answer 6

• Split the integral into two integrals ∫_-∞⁰ +
  ∫₀^∞

Question 7

What is the formula for the mean value of an integral?

Answer 7

1/(b-a) ∫_a^b f(x) dx

Question 8

If the mean value of f(x) is f then what is the mean value of:

f(x) + k ?
kf(x) ?
-f(x) ?

Answer 8

f(x) + k = f +k
kf(x) = kf
-f(x) = -f

Question 9

What are the formulas for the derivatives of inverse trig functions that are given in the formula book?

Answer 9

d/dx (arcsin(x)) = 1 / (1-x²)^1/2
d/dx (arcsin(x)) = 1 / (1+x²)^1/2
d/dx (arcsin(x)) = 1 / 1+x²

Note you must apply the chain rule aswell

Question 10

How do you prove the derivatives of arcsin(x) and arcos(x) ?

Answer 10

• y = inverse trig functions
• x = trig function
• find dx/dy
• find dy/dx
• use the formula for cos²(y) + sin²(y) = 1
• Sub in for x

Question 11

How do you prove the derivative of arctan(x) ?

Answer 11

• y = arctan(x)
• x = tan(y)
• dx/dy = sec²(y)
• dy/dx = 1/sec²(y)
• Use the identity 1 + tan²(y) = sec²(y)
• Sub in for x

Question 12

What are the steps for integrating inverse trig functions?

Answer 12

• let x = asin(u) or atan(u) (u term is the inverse of this so u=arcsin(x/a) or u=arctan(x/a))
• find dx/du rearrange for dx
• replace dx and x in the integral
• rearrange and then use the identity, cos²(u) + sin²(u) = 1
• rearrange and simplify
• swap your limits and solve

Question 13

How do you pick your ‘x’ term in integrating via a trig substitution?

Answer 13

If there is a - on the bottom x=asin(u) if there is a + on the bottom (and generally a root) use x=acos(u)

Question 14

What is the formula for integrating ∫1/(a²-x²)^1/2 that’s given in the formula book?

Answer 14

arcsin(x/a) + c |x|< a

This can be proven using the substitution x=asin(u)

Note you need to apply the reverse chain rule aswell as these formulas

Question 15

What is the formula for integrating ∫1/a²+x² that’s given in the formula book?

Answer 15

1/a arctan(x/a) + c

This can be proven using the substitution x=atan(u)
Note you may need to apply the reverse chain rule

Question 16

When doing a trigonometric substitution what must you keep in mind?

Answer 16

Whatever is on the bottom of the fraction in the integral must have distinct x and a terms.

i.e if you have 4(x+1)² inside the square root you must absorb the 4. Or if you have a quadratic inside the square root you must complete the square

Question 17

How do you integrate inverse trig functions e.g ∫arcsin(x)

Answer 17

• Use parts by integrating 1x(arcsin(x) where arcsin(x) is your ‘u’ term

Question 18

What is the formula for ∫1/(a²-x²) thats given in the formula book?

Answer 18

1/2a ln|(a+x)/(a-x)| + c or 1/a arctanh(x/a) + c
|x|<a

Note you need to apply the chain rule aswell as this formula

This formula can be proven by rewriting it interms of partial fractions and using ln integration. Then using the substitution x=atanh(u)

Question 19

What is the formula for ∫1/(x²-a²)

Answer 19

1/2a ln|(x-a)/(x+a)|+ c

Note you may need to apply the chain rule aswell

This formula can be proven by rewriting it interms of partial fractions and using ln integration

Question 20

What do you do in partial fractions if the top and the bottom of the fraction have a term with an equal highest power of x (before factorising) ? e.g x⁴+x / x⁴+5x²+6

Answer 20

In your form for the partial fractions you will have a term that is just constant

Question 20

What do you do in partial fractions when the bottom has a quadratic term? e.g
1+x / x(x²+9)

Answer 20

In your form for the partial fractions the terms that have a quadratic on the bottom are in the form A+Bx on the top not just A

Question 21

In complicated partial fractions what is an alternative to subbing in values of x?

Answer 21

Comparing coefficients on the LHS and RHS