Answer to Question #5791 in Geometry for maria prettymaths
int(f*g)dx = f*int(g)dx - int(f'*int(g)dx)dx.
Note that on the right-hand side, ƒ is differentiated and g is integrated; consequently it is useful to choose ƒ as a function that simplifies when differentiated, and/or to choose g as a function that simplifies when integrated. As a simple example, consider:
Since the derivative of ln x is 1/x, we make this part of ƒ; since the anti-derivative of 1/x2 is −1/x, we make this part of g. The formula now yields:
int(ln(x)/x²)dx = -ln(x)/x - int((1/x)(-1/x))dx.
The remaining integral of −1/x2 can be completed with the power rule and is 1/x.
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