Question Number 8734 by tawakalitu last updated on 24/Oct/16
![Evaluate the integral ∫[a(b^• .a + b.a^• ) + a^• (b.a) − 2(a^• .a)b − b^• ∣a∣^2 ]dt In which a^• ,b^• are the derivatives of a,b with respect to t](Q8734.png)
$$\mathrm{Evaluate}\:\mathrm{the}\:\mathrm{integral} \\ $$$$\int\left[\mathrm{a}\left(\overset{\bullet} {\mathrm{b}}.\mathrm{a}\:+\:\mathrm{b}.\overset{\bullet} {\mathrm{a}}\right)\:+\:\overset{\bullet} {\mathrm{a}}\left(\mathrm{b}.\mathrm{a}\right)\:−\:\mathrm{2}\left(\overset{\bullet} {\mathrm{a}}.\mathrm{a}\right)\mathrm{b}\:−\:\overset{\bullet} {\mathrm{b}}\mid\mathrm{a}\mid^{\mathrm{2}} \right]\mathrm{dt} \\ $$$$\mathrm{In}\:\mathrm{which}\:\:\overset{\bullet} {\mathrm{a}}\:,\overset{\bullet} {\mathrm{b}}\:\:\mathrm{are}\:\mathrm{the}\:\mathrm{derivatives}\:\mathrm{of}\:\:\mathrm{a},\mathrm{b}\:\mathrm{with}\: \\ $$$$\mathrm{respect}\:\mathrm{to}\:\mathrm{t} \\ $$
Commented by Yozzias last updated on 24/Oct/16
$$\int\left(\mathrm{a}\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{a}.\mathrm{b}\right)−\overset{\bullet} {\mathrm{a}}\left(\mathrm{a}.\mathrm{b}\right)−\overset{\bullet} {\mathrm{b}}\left(\mathrm{a}^{\mathrm{2}} \right)\right)\mathrm{dt} \\ $$$$\int\left(\mathrm{a}\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{a}.\mathrm{b}\right)−\mathrm{a}\left\{\overset{\bullet} {\mathrm{a}}\left(\mathrm{b}\right)+\overset{\bullet} {\mathrm{b}}\left(\mathrm{a}\right)\right\}\right)\mathrm{dt} \\ $$$$=\int\left(\mathrm{a}\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{a}.\mathrm{b}\right)−\mathrm{a}\frac{\mathrm{d}}{\mathrm{dt}}\left(\mathrm{a}.\mathrm{b}\right)\right)\mathrm{dt} \\ $$$$=\mathrm{c} \\ $$
Commented by tawakalitu last updated on 25/Oct/16
$$\mathrm{Thank}\:\mathrm{you}\:\mathrm{sir}.\: \\ $$