Calculus II Final Exam Cheat Sheet
✅INTEGRALS (When you see ∫...)
1. Substitution (u-sub)
What to do:
- Pick the inside part as 'u'
- Replace everything in terms of 'u'
- Integrate
- Plug 'x' back in
Example:
∫ x cos(6x² - 1) dx
Let u = 6x² - 1 → du = 12x dx → dx = du/12x
Then: ∫ x cos(u) * (1/12x) du = (1/12) ∫ cos(u) du = (1/12) sin(u) + C
Final Answer: (1/12) sin(6x² - 1) + C
2. Integration by Parts
Formula: ∫ u dv = uv - ∫ v du
What to do:
- Pick u: something that gets simpler when you take the derivative
- Pick dv: something you can integrate
- Do the formula step-by-step
Example:
∫ 2u e^u du → u = 2u, dv = e^u du
Then: du = 2 du, v = e^u
Answer: 2u e^u - 2e^u + C
3. Long Division + Partial Fractions
What to do:
- Divide the polynomials (like regular long division)
- Break the remainder into simple fractions
- Integrate each piece
Example:
∫ (x³ - 8x² + 21x - 19)/(x² - 6x + 9) dx
Divide → x - 2 + (-1)/(x - 3)²
Answer: (x - 2)²/2 + 1/(x - 3) + C
✅LIMITS (No L’Hôpital Rule)
If the limit gives 0/0 and you can’t use L’Hôpital:
- Use Taylor expansions or series approximation:
• sin(x) ≈ x - x³/6
• tan(x) ≈ x + x³/3
✅TAYLOR POLYNOMIAL & SERIES
1. What is a Taylor Polynomial?
Formula:
Pₙ (x) = f(a) + f'(a)(x - a) + f''(a)/2!(x - a)² + ... + fⁿ(a)/n!(x - a)ⁿ
What to do:
- Pick a good point like a = 1
- Take derivatives up to needed degree
- Plug in and build the polynomial
2. Radius of Convergence
Use the Ratio Test:
R = 1 / limit as n→∞ of |aₙ ₊₁ / aₙ |
3. Error Term Estimate (Remainder)
Formula:
Error ≤ |x - a|ⁿ⁺¹ / (n+1)! * max |f⁽ⁿ⁺¹⁾(t)|
Simple version: Error gets smaller as n gets bigger!
✅QUICK DERIVATIVES / INTEGRALS TO MEMORIZE
Function
Derivative
Integral
xⁿ
n xⁿ⁻¹
xⁿ⁺¹ / (n+1) + C
e^x
e^x
e^x + C
sin x
cos x
-cos x + C
cos x
-sin x
sin x + C
ln x
1/x
x ln x - x + C
√x
1 / (2√x)
(2/3)x^(3/2) + C