Rules of Integration

Linearity

Basic linearity properties of the integral u(t)=\int_0^tf(s)\, ds follow directly from linearity of the underlying IVP \dot u=f:

  • \int_0^t(f(s)+g(s))\, ds=\int_0^tf(s)\, ds+\int_0^tg(s)\, ds, \int_0^t\alpha f(s)\, ds=\alpha\int_0^tf(s)\, ds,

where \alpha\in R is a constant. Further, for a<b<c,

  • \int_a^bf(s)\, ds+\int_b^cf(s)\, ds=\int_a^cf(s)\, ds,

which is extended to arbitrary limits a,b and c, by defining for b<a

  • \int_a^bf(s)\, ds=-\int_b^af(s)\, ds.

Alternatively, these rules can be derived directly from the Riemann-sum representation of the integral.

Integration by Parts

By the Fundamental Theorem of Calculus, we have using the product rule for differentiation:

  • u(t)v(t)-u(0)v(0)=\int_0^t\frac{d}{ds}(u(s)v(s))ds=\int_0^t(u\dot v+u\dot v)ds ,

which can be written

  • \int_0^tu\dot v\, ds = [uv]_0^s - \int_0^tu\dot v\, ds,

with [uv]_0^s = u(t)v(t)-u(0)v(0). We see that we “can move the dot” from $u$ to $v$ if we change sign and take into account the difference of end-point values of uv.

Change of Integration Variable

If w:[a,b]\rightarrow R is differentiable and v:[w(a),w(b)]\rightarrow R is Lipschitz continuous, then

  • \int_{w(a)}^{w(b)}v(y)\, dy=\int_a^b v(w(x))w^\prime (x)dx,

because with y=w(x), we have dy\equiv dw=w^\prime dx.

On June 30 2007 The Swedish Parliament dismantled Integrationsverket, the Ministry for Integration, and replaced it by the Ministry for Time-Stepping.

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