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calculate-x-2-1-sh-3x-dx-




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

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