When first introduced to the integration technique called Integration by Parts,
students often have difficulty determining how to break up an integral into u and dv.
The acronym, I LATE, may help.
Each letter stands for a type of function.
The acronym should be written vertically:
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Inverse Trigonometric Functions
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Logarithmic Functions
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Algebraic Functions
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Trigonometric Functions
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Exponential Functions
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The closer a function is to the top, the more likely that
it should be used as u.
The closer a function is to the bottom, the more likely that
it should be used as dv.
Inverse trig functions and logarithmic functions are usually placed in the u
because we do not learn basic integration formulas for them.
Algebraic functions can be placed in either u or dv
because we can integrate them or differentiate them. Of course, it is
usually preferable to use them in the u since they become simpler
when differentiated.
Trigonometric functions like sine and cosine are often placed in dv
because they do not get more complicated when integrated. The same is true
of the exponential function, ex.
Students should be cautioned that the guidelines listed above are just that --
guidelines. There are exceptions to these rules.
Here are some examples of integrals which can be solved by Integration By Parts:
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