Integral Calculus

What is the integration by parts (IBP) theorem?

If u, v : [a, b] → R are differentiable and u’, v’ are integrable then ^b_a∫ uv’ = [uv]^b_a - ^b_a∫ u’v (for antiderivatives use ∫ uv’ = uv - ∫ u’v)

What is the substitution theorem?

If g : [a, b] → R is differentiable and g’ is integrable and f : R → R is continuous, then ^b_a∫ f(g(x))g’(x)dx = ^g(b)_g(a)∫ f(u)du

What is the method of partial fractions (∫ P(x)/Q(x) where P, Q are polynomials)?

1. If deg P >/ deg Q, then use long division

2. Factorise Q(x) into linear/quadratic

3. Choose correct partial fraction expansion

4. Integrate

What are the partial fraction expansions for different factors?

What are some useful antiderivatives?

• ∫ 1/x dx = log|x|

• ∫ 1/(x² + a²) dx = 1/a(tan^-1(x/a))

• ∫ f’(x)/f(x) dx = log |f(x)|

What is the method for integrating ∫sin^m(x)coxⁿ(x) dx when n is odd?

• Take out a cos(x) (e.g. ∫sin²(x)cos³(x) dx = ∫sin²(x)cos²(x)(cos(x) dx)

• Let u = sinx so du = cos(x) dx and use cos²(x) = 1 - sin²(x)

What is the method for integrating ∫sin^m(x)coxⁿ(x) dx when m is odd?

• Suppose m = 2k + 1 where k = 0, 1,…

• Take out a sin(x) (e.g. ∫sin^2k+1(x)cosⁿ(x) dx = ∫sin^2k(x)cosⁿ(x)(sin(x) dx)

• Let u = cosx so du = -sin(x) dx and use sin^2k(x) = (sin²x)^k = (1 - cos²(x))^k = (1 - u²)^k

What is the method for integrating ∫sin^m(x)coxⁿ(x) dx when m and n are even?

Use

• sin²x = 1/2(1 - cos(2x))

• cos²x = 1/2(1 + cos(2x))

• sin(x)cos(x) = 1/2sin(2x)

What are some important trig identities and derivatives involving tan and sec?

• tan²x + 1 = sec²x

• d/dx(tanx) = sec²x

• d/dx(secx) = sec(x)tan(x)

What is the method for integrating ∫tan^m(x)secⁿ(x) dx when m is even?

• Let n = 2k, u = tanx so du = sec²x

• So ∫tan^m(x)sec^2k(x) dx = ∫u^m(u² + 1)^k-1 du

What is the method for integrating ∫tan^m(x)secⁿ(x) dx when m is odd?

• Let m = 2k + 1, u = secx and du = sec(x)tan(x)

• Then ∫tan^2k+1(x)secⁿ(x) dx = ∫tan^2k(x)sec^n-1(x)(tan(x)sec(x)dx) = ∫(u² - 1)^ku^n-1 du

What is the trigonometric substitution for √(a² - x²)?

x = asinθ or x = acosθ

What is the trigonometric substitution for √(a² + x²)?

x = atanθ (using tan²θ + 1 = sec²θ)

What is the trigonometric substitution for √(x² - a²)?

x = asecθ (using tan²θ + 1 = sec²θ)