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Question Number 70446 by oyemi kemewari last updated on 04/Oct/19

Commented by oyemi kemewari last updated on 04/Oct/19

please help me solve this question

Commented by mathmax by abdo last updated on 04/Oct/19

$$itsonlyatryletf\left(x\right)={\int }_0^x\sqrt{x-t}sintdtchangement\sqrt{x-t}=ugive$$$$x-t=u^2\Rightarrow t=x-u^2\Rightarrow f\left(x\right)=-{\int }_0^{\sqrt{x}}usin(x-u^2)(-2u)du$$$$=2{\int }_0^{\sqrt{x}}{u}^{2}sin(x-u^2)du={\int }_0^{\sqrt{x}}u\left(2usin(x-u^2)\right)du$$$$bypartsf=uand{g}^{{}^{\prime }}=2usin(x-u^2)\Rightarrow$$$$f\left(x\right)={\left[ucos(x-u^2)\right]}_0^{\sqrt{x}}-{\int }_0^{\sqrt{x}}cos(x-u^2)du$$$$=\sqrt{x}cos(\sqrt{x}-x)-{\int }_0^{\sqrt{x}}(cosxcos\left(u^2\right)+sinxsin\left(u^2\right)du$$$$=\sqrt{x}cos(\sqrt{x}-x)-cosx{\int }_0^{\sqrt{x}}cos\left(u^2\right)du-sinx{\int }_0^{\sqrt{x}}sin\left(u^2\right)du$$

Commented by MJS last updated on 04/Oct/19

$$\mathrm{this}\mathrm{looks}\mathrm{as}\mathrm{if}{\mathrm{it}}^{\prime }\mathrm{s}\mathrm{coming}\mathrm{to}\mathrm{a}\mathrm{Fresnel}\mathrm{integral},$$$$\mathrm{but}\mathrm{I}\mathrm{cannot}\mathrm{help}...$$

Commented by mathmax by abdo last updated on 04/Oct/19

$$youarerightsir.$$

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