Integration by Parts: Example 1

Evaluate

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Let us follow the steps

1
This is an indefinite integral involving one function. The second needed function is g(x) = 1. Since the derivative of this function is 0, the only choice left is to differentiate the other function tex2html_wrap_inline30 .
2
We have

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which gives

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3
We have the formula

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4
Since

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we get

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Since the derivative of tex2html_wrap_inline42 is 1/x, it is very common that whenever an integral involves a function which is a product of tex2html_wrap_inline42 with another function, to differentiate tex2html_wrap_inline42 and integrate the other function.

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