Question Number 224948 by fantastic last updated on 13/Oct/25
$$\int\:\sqrt{{x}}\mathrm{sin}\:{x}\:{dx} \\ $$
Commented by Tawa11 last updated on 13/Oct/25
$$\mathrm{Too}\:\mathrm{tiny}. \\ $$$$\:\:\int\:\sqrt{\mathrm{x}}\:\mathrm{sin}\left(\mathrm{x}\right)\:\mathrm{dx} \\ $$
Answered by Frix last updated on 13/Oct/25
![∫(√x)sin x dx =^([by parts]) −(√x)cos x +(1/2)∫((cos x)/( (√x)))dx (1/2)∫((cos x)/( (√x)))dx =^([t=(√((2x)/π))]) ((√(2π))/2)∫cos ((πt^2 )/2) dt = (This is the Fresnel Integral) =((√(2π))/2)C (t) ⇒ ∫(√x)sin x dx=((√(2π))/2)C (((√(2x))/( (√π)))) −(√x)cos x +C](https://www.tinkutara.com/question/Q224955.png)
$$\int\sqrt{{x}}\mathrm{sin}\:{x}\:{dx}\:\overset{\left[\mathrm{by}\:\mathrm{parts}\right]} {=} \\ $$$$−\sqrt{{x}}\mathrm{cos}\:{x}\:+\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{cos}\:{x}}{\:\sqrt{{x}}}{dx} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{\mathrm{cos}\:{x}}{\:\sqrt{{x}}}{dx}\:\overset{\left[{t}=\sqrt{\frac{\mathrm{2}{x}}{\pi}}\right]} {=}\:\frac{\sqrt{\mathrm{2}\pi}}{\mathrm{2}}\int\mathrm{cos}\:\frac{\pi{t}^{\mathrm{2}} }{\mathrm{2}}\:{dt}\:= \\ $$$$\:\:\:\:\:\left(\mathrm{This}\:\mathrm{is}\:\mathrm{the}\:\mathrm{Fresnel}\:\mathrm{Integral}\right) \\ $$$$=\frac{\sqrt{\mathrm{2}\pi}}{\mathrm{2}}\mathrm{C}\:\left({t}\right) \\ $$$$\Rightarrow \\ $$$$\int\sqrt{{x}}\mathrm{sin}\:{x}\:{dx}=\frac{\sqrt{\mathrm{2}\pi}}{\mathrm{2}}\mathrm{C}\:\left(\frac{\sqrt{\mathrm{2}{x}}}{\:\sqrt{\pi}}\right)\:−\sqrt{{x}}\mathrm{cos}\:{x}\:+{C} \\ $$
Commented by fantastic last updated on 13/Oct/25
$${Wait}\:{isnt}\:{this}\:{non}\:{integrateble}… \\ $$
Commented by Frix last updated on 13/Oct/25
$$\mathrm{As}\:\mathrm{you}\:\mathrm{can}\:\mathrm{see}\:\mathrm{it}'\mathrm{s}\:\mathrm{possible}. \\ $$
Commented by fantastic last updated on 13/Oct/25
$${yeah}.. \\ $$
Commented by fantastic last updated on 13/Oct/25
$${Q}\mathrm{224926} \\ $$
Answered by Ghisom_ last updated on 13/Oct/25
$$\int\sqrt{{x}}\:\mathrm{sin}\:{x}\:{dx}=\frac{\mathrm{i}}{\mathrm{2}}\int\mathrm{e}^{−\mathrm{i}{x}} \sqrt{{x}}\:{dx}−\frac{\mathrm{i}}{\mathrm{2}}\int\mathrm{e}^{\mathrm{i}{x}} \sqrt{{x}}\:{dx} \\ $$$$\mathrm{knowing}\:\Gamma\:\left({a},\:{bx}\right)\:=−\int\frac{\mathrm{e}^{−{bx}} \left({bx}\right)^{{a}} }{{x}}{dx}\:\mathrm{we}\:\mathrm{get} \\ $$$$\int\sqrt{{x}}\:\mathrm{sin}\:{x}\:{dx}=−\sqrt{\mathrm{i}}\left(\Gamma\:\left(\frac{\mathrm{3}}{\mathrm{2}},\:−\mathrm{i}{x}\right)\:−\mathrm{i}\Gamma\:\left(\frac{\mathrm{3}}{\mathrm{2}},\:\mathrm{i}{x}\right)\right)\:+{C} \\ $$