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Unit 3: Integration by Parts

The product rule run in reverse, trading one hard integral for an easier one plus a boundary term.

5 topics · 20 questions

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Integration by Parts as the Product Rule in Reverse

  1. How it worksHow does integrating both sides of the product rule d/dx(uv) = u'v + uv' produce the integration by parts formula?
  2. How it worksHow does the integral of u dv turning into uv minus the integral of v du look like two areas rearranging?
  3. Worked exampleFind the integral of x times ex dx using integration by parts.
  4. Common mistakeWhy doesn't the integral of f(x) times g(x) dx equal the integral of f(x) dx times the integral of g(x) dx?
  5. Going furtherWhat happens to the integration by parts formula when u and dv are swapped from the better choice?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4x ex, satisfying F(0) = 0.

Choosing u and dv

  1. How it worksHow does the LIATE order help you pick which factor becomes u before you even start?
  2. How it worksHow does trying dv = x dx instead of dv = ln(x) dx on the integral of x ln(x) dx reveal the wrong choice quickly?
  3. Worked exampleFind the integral of x2 times ln(x) dx by choosing u = ln(x).
  4. Common mistakeWhy doesn't choosing u as the factor that's easiest to integrate usually work out?
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4x2 ln(x), satisfying F(1) = 0.

Repeated Integration by Parts and the Tabular Method

  1. How it worksHow does the tabular method turn several rounds of integration by parts into one alternating-sign table?
  2. Common mistakeWhy doesn't the tabular method stop working once a derivative column reaches zero?
  3. Worked exampleFind the integral of x2 times ex dx using the tabular method.
  4. Worked exampleFind the integral of x3 times sin(x) dx using the tabular method.
Your turn1 of 2
Find F(x), the antiderivative of f(x) = 4x2 ex, satisfying F(0) = 0.

Integration by Parts on a Definite Integral

  1. How it worksHow does evaluating uv at the bounds before integrating v du keep a definite integral by parts organized?
  2. Worked exampleFind the definite integral from 0 to 1 of x times ex dx using integration by parts.
  3. Common mistakeWhy isn't it safe to integrate by parts first and only plug in the bounds at the very end?
  4. Going furtherWhat happens to the boundary term when u or v vanishes at both endpoints?
Your turn1 of 2
Find the definite integral from 0 to 2 of 3x ex dx using integration by parts.

Reduction Formulas

  1. How it worksHow does applying integration by parts once to sinn(x) produce a reduction formula in n minus 2?
  2. Worked exampleFind the integral of sin4(x) dx using the reduction formula for sinn(x).
  3. Common mistakeWhy doesn't a reduction formula solve an integral in one step instead of stepping the power down repeatedly?
Your turn1 of 2
Find the definite integral from 0 to π/2 of 4 sin4(x) dx.