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Question Number 137378 by benjo_mathlover last updated on 02/Apr/21

$$Given{\underset{0}{\int }}^{x}f\left(t\right)dt=x^2\sin \left(\pi x\right)$$$$findf\left(2\right).$$

Answered by EDWIN88 last updated on 02/Apr/21

$$\frac{\mathrm{d}}{\mathrm{dx}}\left[{\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}\right]=\frac{\mathrm{d}}{\mathrm{dx}}\left[{\mathrm{x}}^2\sin \left(\pi \mathrm{x}\right)\right]$$$$\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=2\mathrm{x}\sin \left(\pi \mathrm{x}\right)+\pi {\mathrm{x}}^2\cos \left(\pi \mathrm{x}\right)$$$$\Rightarrow \mathrm{f}\left(2\right)=4\sin \left(2\pi \right)+4\pi \cos \left(2\pi \right)=4\pi$$

Answered by mathmax by abdo last updated on 02/Apr/21

$${\int }_0^{\mathrm{x}}\mathrm{f}\left(\mathrm{t}\right)\mathrm{dt}={\mathrm{x}}^2\sin \left(\pi \mathrm{x}\right)\Rightarrow \mathrm{f}\left(\mathrm{x}\right)=2\mathrm{xsin}\left(\pi \mathrm{x}\right)+\pi {\mathrm{x}}^2\cos \left(\pi \mathrm{x}\right)\Rightarrow$$$$\mathrm{f}\left(2\right)=4\sin \left(2\pi \right)+4\pi \cos \left(2\pi \right)=4\pi$$

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