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Last modified: January 26, 2018
Integration By Parts
--------------------
Integration of the product rule leads to:
∫udv = uv - ∫vdu
Example 1: ∫xcosxdx
u = x ∴ du = dx
dv = cosxdx ∴ v = sinx
∫xcosx = ∫udv
= uv - ∫vdu
= xsinx - ∫sinxdx
= xsinx = cosx + C
Example 2: ∫ln(3 - x)dx
u = ln(3 - x) ∴ du = -dx/(3 - x) = dx/(x - 3)
v = (x - 3) ∴ dv = dx
∫ln(3 - x) = ∫udv
= uv - ∫vdu
= (x - 3)ln(3 - x) - ∫(x - 3)dx/(x - 3)
= (x - 3)ln(3 - x) + x + C
Example 3: ∫xexp(x)dx
u = x ∴ du = dx
v = exp(x) ∴ dv = exp(x)dx
∫xexp(x)dx = ∫udv
= uv - ∫vdu
= xexp(x) - ∫exp(x)dx
= xexp(x) - exp(x) + C