Pure 2.3 - Methods in Calculus

Question 1

What are improper integrals?

Answer 1

The integral ∫ₐᵇf(𝑥)d𝑥 is improper if either:
One or both of the limits is infinite
f(𝑥) is undefined 𝑥=𝑎, 𝑥=𝑏 or another point in the interval [𝑎,𝑏].

Question 2

What does it mean for an improper integral is convergent or divergent.

Answer 2

If an improper integral exists then it is said to be convergent. If it doesn’t exist then it is said to be divergent.

Question 3

How can you work with improper integral where both limits are infinite.

Answer 3

The integral must be split into the sum of two improper integrals.
If both of these integrals converge, then so will the original integral.
If on or both of these integrals diverge, then so will the original integral.

Question 4

If the function f(𝑥) has the mean value f̅ over the interval [𝑎,𝑏], and 𝑘 is a real constant, then what would the mean values of f(𝑥)+𝑘, 𝑘f(𝑥) and -f(𝑥) be over the same interval.

Answer 4

f(𝑥)+𝑘 has mean value f̅+𝑘 over the interval [𝑎,𝑏]
𝑘f(𝑥) has mean value 𝑘f̅ over the interval [𝑎,𝑏]
-f(𝑥) has mean value -f̅ over the interval [𝑎,𝑏]

Question 5

What is the result of (d/d𝑥)(arcsin(𝑥))?

Answer 5

(d/d𝑥)(arcsin(𝑥))=1/√(1-𝑥²)

Question 6

What is the result of (d/d𝑥)(arccos(𝑥))?

Answer 6

(d/d𝑥)(arccos(𝑥)) = -1/√(1-𝑥²)

Question 7

What is the result of (d/d𝑥)(arctan(𝑥))?

Answer 7

(d/d𝑥)(arctan(𝑥)) = 1/(1+𝑥²)

Question 8

What is the result of ∫(1/(𝑎²+𝑥²))d𝑥?

Answer 8

∫(1/(𝑎²+𝑥²))d𝑥=(1/𝑎)arctan(𝑥/𝑎)+𝑐

Question 9

What is the result of ∫(1/√(𝑎²-𝑥²))d𝑥?

Answer 9

∫(1/√(𝑎²-𝑥²))d𝑥=arcsin(𝑥/𝑎)+𝑐