Calculus
Integration by Parts
The product rule run backwards, as a picture. Plot one factor against the other and the curve cuts a rectangle into two pieces: one is the integral you were asked for, the other is the one you are left with, and they must add up to the corners. Choosing u is choosing which you would rather do.
∫ u dv, the one you wanted∫ v du, the one you are left with[uv], the corners
Every number is integrated numerically from the two functions, so the identity is checked on screen
The integral you wanted—
—
The integral
Call this one u
a few wide onestoo fine to see
Values
u at b—
v at b—
∫ u dv—
∫ v du—
[uv] from a to b—
the two minus [uv]—
the strips on screen—
strips minus [uv]—
What to observe
- The curve (u, v) cuts the corner in two: the area under it is ∫v du, the area beside it is ∫u dv, and together they fill exactly[uv].
- Call eˣ u in ∫x eˣ dx and the leftover becomes ∫(x²/2)eˣ dx, worse than where you started. Choosing u is choosing the leftover.
- ∫ ln x dx has only one factor, but dv = dx is always available: the leftover is ∫1 dx.
- On ∫ x² eˣ dx the leftover ∫2x eˣ dx is one power lighter. Repeat the move and it is gone: by parts is often a staircase.
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