(BC only) Integration by parts, partial fractions and improper integrals

Learning objectives

Apply integration by parts in the form ∫u dv = uv − ∫v du, choosing u so that differentiating it simplifies the integrand, and apply the technique to products such as x eˣ, x sin x and ln x

Use integration by parts repeatedly, and solve for the original integral when the same integral reappears on the right, as happens with ∫eˣ sin x dx

Decompose a rational function whose denominator factors into distinct non-repeating linear factors into partial fractions, and integrate each term to obtain a sum of logarithms

Identify an integral as improper because a limit of integration is infinite or because the integrand is unbounded at an endpoint, and rewrite it with a limit before evaluating

Evaluate an improper integral by computing the limit, and state clearly whether the integral converges and to what value, or diverges

Apply improper integrals to an accumulation continuing without end, such as the total amount produced by a decaying rate, and interpret the finite total when it exists