## Integration by Parts ... the LIATE heuristic

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### Integration by Parts ... the LIATE heuristic

When you apply integration by parts, there is usually a choice of what to call u and what to call dv. The "LIATE" heuristic provides a suggestion of how to do that. It doesn't always work, but it works so often that it is worth remembering and using it as the first attempt.

LIATE stands for:
Logarithmic
Inverse trigonometric
Algebraic
Trigonometric
Exponential

The heuristic says: when there are two factors in the integrand, choose u as the one furthest to the left in LIATE and choose dv as the one furthest to the right.

Example... $\int x^2 \,e^{-x}\,dx$ ... Here $x^2$ is algebraic, $e^{-x}$ is exponential. So use $u=x^2, dv = e^{-x}dx$

$\int z^2\, \ln z\,dz$ ... Here $z^2$ is algebraic, $\ln z$ is logarithmic. So use $u=\ln z, dv=z^2dz$

Try this... $\int e^{2x} \arcsin(5x)\,dx$

g_edgar
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