Integration by Parts

Recall that du = f(t)*dt and dv = g(t)*dt in Euler forward form expresses that

  • u(t+dt) = u(t) + f(t)*dt    or   du = u(t+dt) – u(t) = f(t)*dt
  • v(t+dt) = v(t) + g(t)*dt    or   dv = v(t+dt) – v(t) = g(t)*dt

Consider now d(u*v) in the form

  • d(u*v) = u(t+dt)*v(t+dt) – u(t)*v(t)
  • = u(t+dt)*(v(t+dt)- v(t)) + (u(t+dt)-u(t))*v(t)
  • = u(t+dt)*dv + du*v(t)

which shows that for small dt

  • d(u*v) = u(t)*dv + v(t)*du = u(t)*g(t)*dt + v(t)*f(t)*dt

We can express this relation in two ways, as derivative of a product u*v:

  • d(u*v)/dt= u*dv/dt + v*du/dt = u*g + v*f   (with g = dv/dt and f = du/dt)

or by integration from 0 to T,

  • u(T)*v(T) – u(0)*v(0) = ∫ (u*g + v*f)*dt    (with g = dv/dt and f = du/dt)

which we can write in the form

  • ∫ u*dv/dt*dt = u(T)*v(T) – u(0)*v(0) – ∫ v*du/dt*dt

referred to as integrations by parts (shifting the derivative from v to u.

The formula for the derivative of a product and the related formula for integration by parts are the corner stones of Symbolic Calculus filling traditional books of Calculus page up and down.