Integrals · Definition

Integration by parts

calculusintegrals

Before you start you need to know

Technique that reverses the product rule to integrate products of functions.

Definition

Integration by parts is a technique for integrating products of functions, derived from the product rule. It transforms \(\int u\,dv\) into \(uv-\int v\,du\).

\[\int u\,dv=uv-\int v\,du\]

Integration by parts is the product rule in reverse: \(\int u\,dv=uv-\int v\,du\). It is used when the integrand is a product of two kinds of function (for example, a polynomial times an exponential).

The key is choosing \(u\) well (what simplifies when differentiated) and \(dv\) (what you know how to integrate). A useful order: logarithmic, inverse trig, algebraic, trigonometric, exponential (LIATE).

Sometimes you must apply it twice, and in some cases the original integral reappears and can be solved for.

Think about it

Compute \(\int x\,e^x\,dx\).

With \(u=x\), \(dv=e^x\,dx\): \(xe^x-e^x+C\).

Exercises 10

Compute the following integrals

  1. \(\displaystyle \int x e^{x}\,dx\)
  2. \(\displaystyle \int x \sin {x}\,dx\)
  3. \(\displaystyle \int \ln{\left(x \right)}\,dx\)
  4. \(\displaystyle \int x^{2} e^{x}\,dx\)
  5. \(\displaystyle \int x \cos {x}\,dx\)

Problems

  1. When computing \(\displaystyle\int x\,e^{x}\,dx\) by parts, which choice of u and dv is suitable?
  2. What is the integration by parts formula?
  3. According to the LIATE rule, which family of functions should be chosen first as \(u\)?
  4. Which choice of \(u\) and \(dv\) is suitable for \(\displaystyle\int x\ln x\,dx\)?
  5. If \(u=x\) and \(dv=\cos x\,dx\), what are \(du\) and \(v\)?

References

  • James Stewart, Calculus: Early Transcendentals, 8.ª edición, Cengage Learning, 2015.