Integration by parts
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Integration by Parts
Use the reverse of the product rule to integrate products of functions.
Formula
Integration by parts comes from the product rule.
\[ \int u\,\frac{dv}{dx}\,dx=uv-\int v\,\frac{du}{dx}\,dx \]
Many students remember it as \(\int u\,dv=uv-\int v\,du\).
Choosing \(u\)
A useful order of preference is LIATE: logarithmic, inverse trig, algebraic, trigonometric, exponential.
Worked example
Find \(\int x e^x\,dx\).
- Let \(u=x\), so \(du=dx\).
- Let \(dv=e^x dx\), so \(v=e^x\).
- \(\int x e^x dx=xe^x-\int e^x dx=e^x(x-1)+C\).
Classic example
\[\int \ln x\,dx=x\ln x-x+C\]
Use \(u=\ln x\) and \(dv=dx\).
Integration by parts example
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Integration by parts practice
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