Back Calculus 2 Complete Course Cheat Sheet Math
Download Open
Show description 2,401 chars · Math

Calculus 2 Complete Course Cheat Sheet

Calculus 2 Complete Course Cheat Sheet





Module 1: Integration Techniques
Module 2: Applications & Sequences
Module 3: Series
Module 4: Parametric & Polar
Quick Reference





Calculus 2 Complete Course Cheat Sheet

Integration • Applications • Series • Parametric & Polar Curves





Module 1: Techniques of Integration



Substitution Rule


∫ f(g(x))g'(x) dx = ∫ f(u) du, where u = g(x)




U-Substitution Steps:


Choose u = g(x) (usually the "inside" function)

Find du = g'(x) dx

Substitute u and du into integral

Integrate with respect to u

Substitute back: replace u with g(x)






Example:


∫ 2x cos(x²) dx
Let u = x², then du = 2x dx
∫ cos(u) du = sin(u) + C = sin(x²) + C








Integration by Parts


∫ u dv = uv - ∫ v du
where u and dv are chosen parts of the integrand




LIATE Rule for choosing u:
L - Logarithmic (ln x, log x)
I - Inverse trig (arcsin x, arctan x)
A - Algebraic (x, x², polynomials)
T - Trigonometric (sin x, cos x)
E - Exponential (e^x, a^x)




Integration by Parts Steps:


Choose u (using LIATE) and dv

Find du and v

Apply formula: uv - ∫ v du

Integrate ∫ v du






Example:


∫ x e^x dx
u = x, dv = e^x dx
du = dx, v = e^x
= x e^x - ∫ e^x dx = x e^x - e^x + C = e^x(x-1) + C








Trigonometric Integrals



Powers of sin and cos


∫ sin^m x cos^n x dx

• If m odd: save one sin, convert rest using sin²x = 1-cos²x
• If n odd: save one cos, convert rest using cos²x = 1-sin²x
• If both even: use half-angle formulas






Half-Angle Formulas


sin²x = (1 - cos 2x)/2
cos²x = (1 + cos 2x)/2
sin x cos x = (sin 2x)/2






Powers of tan and sec


∫ tan^m x sec^n x dx

• If n even: save sec²x, convert rest using sec²x = 1+tan²x
• If m odd: save sec x tan x, convert rest








Trigonometric Substitution



√(a² - x²)


x = a sin θ
dx = a cos θ dθ
√(a² - x²) = a cos θ






√(a² + x²)


x = a tan θ
dx = a sec² θ dθ
√(a² + x²) = a sec θ






√(x² - a²)


x = a sec θ
dx = a sec θ tan θ dθ
√(x² - a²) = a tan θ












Partial Fractions


For rational functions P(x)/Q(x) where degree(P) < degree(Q)





Linear Factors


1/((x-a)(x-b)) = A/(x-a) + B/(x-b)






Repeated Linear


1/(x-a)² = A/(x-a) + B/(x-a)²






Quadratic Factors


1/((x²+1)(x-a)) = (Ax+B)/(…