Question Number 76193 by abdomathmax last updated on 25/Dec/19
$${calculate}\:\int\:\:\left({x}^{\mathrm{2}} −\mathrm{1}\right){sh}\left(\mathrm{3}{x}\right){dx} \\ $$
Commented by benjo last updated on 25/Dec/19
$$\mathrm{sir}\:\:\mathrm{do}\:\mathrm{you}\:\mathrm{mean}\:\mathrm{sh}\left(\mathrm{3x}\right)\:=\mathrm{sinh}\:\left(\mathrm{3x}\right)? \\ $$
Commented by mathmax by abdo last updated on 25/Dec/19
$${yes} \\ $$
Commented by abdomathmax last updated on 25/Dec/19
$${let}\:{I}\:=\int\left({x}^{\mathrm{2}} −\mathrm{1}\right){sh}\left(\mathrm{3}{x}\right){dx}\:\:{by}\:{parts} \\ $$$${I}\:=\:\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\mathrm{3}}{ch}\left(\mathrm{3}{x}\right)−\frac{\mathrm{1}}{\mathrm{3}}\int\:\:\left(\mathrm{2}{x}−\mathrm{1}\right){ch}\left(\mathrm{3}{x}\right){dx} \\ $$$${again}\:{by}\:{parts}\:\:\int\left(\mathrm{2}{x}−\mathrm{1}\right){ch}\left(\mathrm{3}{x}\right){dx} \\ $$$$=\frac{\mathrm{2}{x}−\mathrm{1}}{\mathrm{3}}{sh}\left(\mathrm{3}{x}\right)−\frac{\mathrm{2}}{\mathrm{3}}\int\:\:{sh}\left(\mathrm{3}{x}\right){dx} \\ $$$$=\frac{\mathrm{2}{x}−\mathrm{1}}{\mathrm{3}}{sh}\left(\mathrm{3}{x}\right)−\frac{\mathrm{2}}{\mathrm{9}}{ch}\left(\mathrm{3}{x}\right)\:+{c}\:\Rightarrow \\ $$$${I}\:=\frac{{x}^{\mathrm{2}} −\mathrm{1}}{\mathrm{3}}{ch}\left(\mathrm{3}{x}\right)−\frac{\mathrm{2}{x}−\mathrm{1}}{\mathrm{9}}{sh}\left(\mathrm{3}{x}\right)+\frac{\mathrm{2}}{\mathrm{27}}{ch}\left(\mathrm{3}{x}\right)\:+{C}\: \\ $$$$ \\ $$
Commented by benjo last updated on 26/Dec/19
$$\mathrm{sir}\:\mathrm{i}\:\mathrm{think}\:\mathrm{missing}\:\mathrm{in}\:\left(\mathrm{2x}−\mathrm{1}\right)/\mathrm{9} \\ $$
Answered by benjo last updated on 25/Dec/19
$$=\:\left(\mathrm{1}/\mathrm{3}\right)\left(\mathrm{x}^{\mathrm{2}} \:−\mathrm{1}\right)\mathrm{cosh}\left(\mathrm{3x}\right)−\left(\mathrm{2}/\mathrm{9}\right)\mathrm{x} \\ $$$$\mathrm{sinh}\left(\mathrm{3x}\right)+\left(\mathrm{2}/\mathrm{27}\right)\mathrm{cosh}\left(\mathrm{3x}\right)+\mathrm{C} \\ $$