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,

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

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

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

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.