Integration by parts
Integration by Parts
Integration by Parts is a topic in Integration in the California Calculus Standards. It is aligned to Standard 17.0, which requires students to compute integrals using techniques of integration such as integration by parts.
Integration by parts integrates a product using \(\int u\,dv=uv-\int v\,du\), the reverse of the product rule, with the LIATE guide for choosing \(u\).
Theory
Integration by parts integrates a product by trading it for an easier integral: \(\int u\,dv=uv-\int v\,du\). It is the reverse of the product rule, and the LIATE guide tells you which factor to call \(u\).
Integrating the product rule \((uv)'=u'v+uv'\) and rearranging gives the integration by parts formula:
You split the integrand into a part to differentiate (\(u\)) and a part to integrate (\(dv\)). A good choice makes the new integral \(\int v\,du\) simpler than the original.
The LIATE order — Logarithmic, Inverse-trig, Algebraic, Trig, Exponential — ranks which factor to choose as \(u\) (earlier in the list wins).
The formula and the choice guide:
How to integrate by parts
- Choose \(u\) and \(dv\) (use LIATE for \(u\)); the rest is \(dv\).
- Compute \(du\) and \(v\) by differentiating \(u\) and integrating \(dv\).
- Apply \(uv-\int v\,du\) and evaluate the new, simpler integral.
Let \(u=x\), \(dv=e^{x}\,dx\); then \(du=dx\), \(v=e^{x}\). Apply \(uv-\int v\,du\).
| \(\int x e^{x}\,dx\) | \(=\) | \(xe^{x}-\int e^{x}\,dx\) |
| \(=\) | \(xe^{x}-e^{x}+C\) | |
| \(=\) | \((x-1)e^{x}+C\) |
Let \(u=x\), \(dv=\cos x\,dx\); then \(du=dx\), \(v=\sin x\). Apply \(uv-\int v\,du\), using \(\int\sin x\,dx=-\cos x\).
| \(\int x\cos x\,dx\) | \(=\) | \(x\sin x-\int \sin x\,dx\) |
| \(=\) | \(x\sin x-(-\cos x)+C\) | |
| \(=\) | \(x\sin x+\cos x+C\) |
Let \(u=\ln x\), \(dv=dx\); then \(du=\dfrac{1}{x}\,dx\), \(v=x\).
| \(\int \ln x\,dx\) | \(=\) | \(x\ln x-\int 1\,dx\) |
| \(=\) | \(x\ln x-x+C\) |
LIATE ranks logarithmic before algebraic, so \(u=\ln x\).
| \(u\) | \(=\) | \(\ln x,\quad dv=x\,dx\) |
Common pitfalls
Frequently asked questions
What is integration by parts?
A method to integrate a product: \(\int u\,dv=uv-\int v\,du\). It reverses the product rule and trades a hard integral for an easier one.
How do you choose u and dv?
Use LIATE: Logarithmic, Inverse-trig, Algebraic, Trig, Exponential. The factor earliest in this list is \(u\); the rest is \(dv\).
How do you integrate ln x?
By parts with \(u=\ln x\) and \(dv=dx\): the result is \(x\ln x-x+C\).
When do you use integration by parts twice?
When one application still leaves a product, such as \(\int x^2 e^{x}\,dx\). Apply parts again until the remaining integral is elementary.