Partial Fractions & Improper/Discontinuous Integrals

Integration by Parts

∫udv= uv-∫vdu

What does LIPET tell you

The order of priority for choosing u

What is L in LIPET

logs

What is I in LIPET

inverse trig

What is P in LIPET

polynomials

What is E in LIPET

exponentials

What is T in LIPET

trig

How do you apply Partial Differention?

1) If n≥d, start w/ long÷

2) Factor denominator

3) Write equation w/ original integrand on left & fraction for ea. factor on right

4) Solve for all numerators (mult. both sides by LCD)

5) Rewrite original integral & evaluate

What are improper integrals?

Integrals that have ∞

∫a to ∞ f(x)dx

lim b→∞ ∫a to b f(x)dx

∫-∞ to b f(x)dx

lim b→-∞ ∫a to b f(x)dx

∫-∞ to ∞ f(x)dx

∫-∞ to a f(x)dx + ∫a to ∞ f(x)dx

Convergent

real number

divergent

\-∞ & ∞

Discontinuous integrals

integrals with a vertical asymptote

∫a to b f(x)dx

lim t→b⁻ ∫a to t f(x)dx (VA at x=b)

∫a to b f(x)dx

lim t→a+ ∫t to tbf(x)dx (VA at x=a)

∫a to b f(x)dx

∫a to c f(x)dx + ∫c to b f(x)dx (VA is between a&b)

When can we use Tab Method?

When we have an integral that’s the product of a polynomial w/ an exponential, sin or cos func, or e