L16: Integration by Parts Overview

Overview

The transcript introduces integration by parts as a key technique for computing indefinite integrals, derived from the product rule for derivatives. It explains the formula, how to choose u and dv, and works through several examples.

From Product Rule to Integration by Parts

• Product rule: derivative of f(x)g(x) is f′(x)g(x) + f(x)g′(x).
• Rewriting as integrals yields an antiderivative identity.
• Rearranged to isolate one integral and substitute variables.

Integration by Parts Formula

• Derivation with substitutions: u = f(x), du = f′(x) dx; v = g(x), dv = g′(x) dx.
• Final formula: ∫ u dv = uv − ∫ v du.
• Constants of integration can be omitted mid-derivation when both sides have indefinite integrals.

Choosing u and dv: Guidelines

• Pick u so du is computable and preferably simpler than u.
• Everything not chosen as u becomes dv automatically.
• You must be able to find v as an antiderivative of dv.
• Prefer v no more complicated than dv after integration.

Example A: ∫ x e^x dx

• Choice: u = x ⇒ du = 1 dx; dv = e^x dx ⇒ v = e^x.
• Apply formula: ∫ x e^x dx = x e^x − ∫ e^x dx = x e^x − e^x + C.
• Check by differentiating to recover x e^x.

Example B: ∫ x sin x dx

• Choice: u = x ⇒ du = 1 dx; dv = sin x dx ⇒ v = −cos x.
• Apply formula: ∫ x sin x dx = −x cos x + ∫ cos x dx = −x cos x + sin x + C.

Example C: ∫ x ln x dx

• Initial bad choice: u = x, dv = ln x dx fails (v unknown).
• Better choice: u = ln x ⇒ du = (1/x) dx; dv = x dx ⇒ v = x^2/2.
• Apply formula: ∫ x ln x dx = (x^2/2) ln x − ∫ (x^2/2)(1/x) dx.
• Simplify: = (x^2/2) ln x − ∫ (x/2) dx = (x^2/2) ln x − x^2/4 + C.

Example D: ∫ ln x dx

• Rewrite as product: ∫ ln x · 1 dx.
• Choice: u = ln x ⇒ du = (1/x) dx; dv = 1 dx ⇒ v = x.
• Apply formula: ∫ ln x dx = x ln x − ∫ x(1/x) dx = x ln x − ∫ 1 dx.
• Result: x ln x − x + C.

Heuristic for Choosing u (Ranking)

• Inverse functions (e.g., ln x, arctan x) preferred as u; easy to differentiate, harder to integrate.
• Polynomial functions next; differentiation simplifies degree.
• Trig functions and e^x lower priority for u; better as dv since they integrate cleanly.

Key Terms & Definitions

• Product rule: d/dx[f g] = f′g + f g′.
• Antiderivative: a function whose derivative equals the integrand.
• Integration by parts: ∫ u dv = uv − ∫ v du.
• u, dv: chosen parts of the integrand to fit the formula.
• du, v: derivative of u times dx; antiderivative of dv.

Action Items / Next Steps

• Practice selecting u and dv using the ranking and simplification goals.
• Verify results by differentiating obtained antiderivatives.
• Prepare for cases requiring repeated integration by parts.

Summary Table: Examples and Choices