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Units, Constants and Useful Formulas

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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