Integration by Parts
This section looks at Integration by Parts (Calculus).
From the product rule, we can obtain the following formula, which is very useful in integration:
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It is used when integrating the product of two expressions (a and b in the bottom formula). When using this formula to integrate, we say we are "integrating by parts".
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Sometimes you will have to integrate by parts twice (or possibly even more times) before you get an answer.
Example
Find ∫xe-x dx
Integrating by parts (with v = x and du/dx = e-x), we get:
-xe-x - ∫-e-x dx (since ∫e-x dx = -e-x)
= -xe-x - e-x + constant
We can also sometimes use integration by parts when we want to integrate a function that cannot be split into the product of two things. The trick we use in such circumstances is to multiply by 1 and take du/dx = 1.
Example
Find ∫ ln x dx
To integrate this, we use a trick, rewrite the integrand (the expression we are integrating) as 1.lnx . We then let v = ln x and du/dx = 1 .
Hence ∫ ln x dx = x ln x - ∫ x (1/x) dx
= x lnx - ∫ dx
= x lnx - x + constant