1. Methods in calculus

2. FM Methods in Calculus: Improper integrals
KUS objectives
BAT evaluate improper integrals; the mean of a function; Integrate using trig substitution; integrate using partial fractions
Starter: find
5𝑥 3+ 𝑥 2 𝑑𝑥
=5 3+ 𝑥 2 +𝐶
𝑥 2 𝑒 𝑥 𝑑𝑥
= 𝑥 2 −2𝑥+2 𝑒 𝑥 +𝐶
sin 𝑥 cos 𝑥 1+3 𝑠𝑖𝑛 2 𝑥 𝑑𝑥
=𝑙𝑛 1+3 𝑠𝑖𝑛 2 𝑥 +𝐶

3. The integral 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 is improper if
Notes
𝐴𝑟𝑒𝑎= 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥
A definite integral represents the area enclosed by a continuous function 𝑦=𝑓(𝑥), the x-axis and line 𝑥=𝑎 and 𝑥=𝑏 𝑜 𝑦 𝑥
𝑦= 1 𝑥 2 𝑎
(1)
𝐴𝑟𝑒𝑎= 𝑎 ∞ 1 𝑥 2 𝑑𝑥
An Improper integral represents the area where one of the limits is infinite or where the function is not defined at some point
The integral 𝑎 𝑏 𝑓(𝑥) 𝑑𝑥 is improper if
one or both of the limits is infinite
f(x) is undefined at 𝑥=𝑎 or 𝑥=𝑏 or at a point in the interval [a, b] 𝑜 𝑦 𝑥
𝑦= 1 𝑥 3
(2)
𝐴𝑟𝑒𝑎= 𝑥 𝑑𝑥
If the improper integral exists it is said to be convergent. If it does not exist it is said to be divergent

4. WB A1- limits of inifinity Evaluate each improper integral
𝑎) 1 ∞ 1 𝑥 2 𝑑𝑥 𝑏) 1 ∞ 1 𝑥 𝑑𝑥
Use limit notation to rewrite the integral as
𝑙𝑖𝑚 𝑡→∞ 1 𝑡 1 𝑥 2 𝑑𝑥
= 𝑙𝑖𝑚 𝑡→∞ − 1 𝑥 𝑡 1
= 𝑙𝑖𝑚 𝑡→∞ − 1 𝑡 +1
𝑎𝑠 𝑡→∞ , 1 𝑡 →0 =𝟏 b)
𝑙𝑖𝑚 𝑡→∞ 1 𝑡 1 𝑥 𝑑𝑥
= 𝑙𝑖𝑚 𝑡→∞ ln 𝑥 𝑡 1
= 𝑙𝑖𝑚 𝑡→∞ ln 𝑡 − ln 1
𝑎𝑠 𝑡→∞ , ln 𝑡 →∞
𝒅𝒐𝒆𝒔 𝒏𝒐𝒕 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆

5. WB A2 - undefined limits Evaluate each improper integral
𝑎) 𝑥 2 𝑑𝑥 𝑏) 0 2 𝑥 4− 𝑥 2 𝑑𝑥
Use limit notation to rewrite the integral as
𝑙𝑖𝑚 𝑡→0 𝑡 𝑥 2 𝑑𝑥
= 𝑙𝑖𝑚 𝑡→0 − 1 𝑥 1 𝑡
= 𝑙𝑖𝑚 𝑡→0 −1+ 1 𝑡
𝑎𝑠 𝑡→0 , 1 𝑡 →∞
−1+ 1 𝑡 →∞ as 𝑡→0 𝒅𝒐𝒆𝒔 𝒏𝒐𝒕 𝒄𝒐𝒏𝒗𝒆𝒓𝒈𝒆 b)
𝑙𝑖𝑚 𝑡→2 0 𝑡 𝑥 4− 𝑥 2 𝑑𝑥
= 𝑙𝑖𝑚 𝑡→2 − 4− 𝑥 2 𝑡 0
The function is undefined for 𝑥=2
= 𝑙𝑖𝑚 𝑡→2 − 4− 𝑡 2 +− 4− 0 2
𝑎𝑠 𝑡→2 , 4− 𝑡 2 →0
= (0) = 2

6. NOW DO EX 3A WB A3 – both limits are infinity a) Find 𝑥 𝑒 − 𝑥 2 𝑑𝑥
b) Hence show that converges and find its value
−∞ ∞ 𝑥 𝑒 − 𝑥 2 𝑑𝑥
−∞ ∞ 𝑥 𝑒 − 𝑥 2 𝑑𝑥
=− 1 2 𝑒 − 𝑥 2 +C b)
−∞ ∞ 𝑥 𝑒 − 𝑥 2 𝑑𝑥 = −∞ 0 𝑥 𝑒 − 𝑥 2 𝑑𝑥 + 0 ∞ 𝑥 𝑒 − 𝑥 2 𝑑𝑥
Split into two improper integrals
= 𝑙𝑖𝑚 𝑡→∞ − 1 2 𝑒 − 𝑥 −𝑡 𝑙𝑖𝑚 𝑡→∞ − 1 2 𝑒 − 𝑥 2 𝑡 0
= 𝑙𝑖𝑚 𝑡→∞ − 1 2 −− 1 2 𝑒 − 𝑡 𝑙𝑖𝑚 𝑡→∞ − 1 2 𝑒 − 𝑡 2 −− 1 2
𝑎𝑠 𝑡→∞ , 𝑒 − 𝑡 2 → both integrals converge
= − = 0
The areas cancel out
NOW DO EX 3A

7. One thing to improve is –
KUS objectives
BAT evaluate improper integrals; the mean of a function; Integrate using trig substitution; integrate using partial fractions
self-assess
One thing learned is –
One thing to improve is –

8. END