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Ответ дал:
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1) Обозначим интегра через I:
![I=\int 4^{x}\cdot sinx\, dx=\\\\=[\, u=4^{x},\; du=4^{x}\cdot ln4\, dx\; ,dv=sinx\, dx,\; v=-cosx\, ]=\\\\=-4^{x}\cdot cosx+ln4\cdot \int 4^{x}cosx\, dx=\\\\=[\, u=4^{x},\; du=4^{x}\cdot ln4\, dx,\; dv=cosx\, dx,\; v=sinx\, ]=\\\\=-4^{x}\cdot cosx+ln4(4^{x}\cdot sinx-ln4\cdot \int 4^{x}\cdot sinx\, dx)\; ;\\\\I=-4^{x}\cdot cosx+4^{x}\cdot sinx\cdot ln4-ln^24\cdot I\\\\I+ln^24\cdot I=4^{x}(sinx\cdot ln4-cosx)\\\\I(1+ln^24)=4^{x}(sinx\cdot ln4-cosx) I=\int 4^{x}\cdot sinx\, dx=\\\\=[\, u=4^{x},\; du=4^{x}\cdot ln4\, dx\; ,dv=sinx\, dx,\; v=-cosx\, ]=\\\\=-4^{x}\cdot cosx+ln4\cdot \int 4^{x}cosx\, dx=\\\\=[\, u=4^{x},\; du=4^{x}\cdot ln4\, dx,\; dv=cosx\, dx,\; v=sinx\, ]=\\\\=-4^{x}\cdot cosx+ln4(4^{x}\cdot sinx-ln4\cdot \int 4^{x}\cdot sinx\, dx)\; ;\\\\I=-4^{x}\cdot cosx+4^{x}\cdot sinx\cdot ln4-ln^24\cdot I\\\\I+ln^24\cdot I=4^{x}(sinx\cdot ln4-cosx)\\\\I(1+ln^24)=4^{x}(sinx\cdot ln4-cosx)](https://tex.z-dn.net/?f=I%3D%5Cint+4%5E%7Bx%7D%5Ccdot+sinx%5C%2C+dx%3D%5C%5C%5C%5C%3D%5B%5C%2C+u%3D4%5E%7Bx%7D%2C%5C%3B+du%3D4%5E%7Bx%7D%5Ccdot+ln4%5C%2C+dx%5C%3B+%2Cdv%3Dsinx%5C%2C+dx%2C%5C%3B+v%3D-cosx%5C%2C+%5D%3D%5C%5C%5C%5C%3D-4%5E%7Bx%7D%5Ccdot+cosx%2Bln4%5Ccdot+%5Cint+4%5E%7Bx%7Dcosx%5C%2C+dx%3D%5C%5C%5C%5C%3D%5B%5C%2C+u%3D4%5E%7Bx%7D%2C%5C%3B+du%3D4%5E%7Bx%7D%5Ccdot+ln4%5C%2C+dx%2C%5C%3B+dv%3Dcosx%5C%2C+dx%2C%5C%3B+v%3Dsinx%5C%2C+%5D%3D%5C%5C%5C%5C%3D-4%5E%7Bx%7D%5Ccdot+cosx%2Bln4%284%5E%7Bx%7D%5Ccdot+sinx-ln4%5Ccdot+%5Cint+4%5E%7Bx%7D%5Ccdot+sinx%5C%2C+dx%29%5C%3B+%3B%5C%5C%5C%5CI%3D-4%5E%7Bx%7D%5Ccdot+cosx%2B4%5E%7Bx%7D%5Ccdot+sinx%5Ccdot+ln4-ln%5E24%5Ccdot+I%5C%5C%5C%5CI%2Bln%5E24%5Ccdot+I%3D4%5E%7Bx%7D%28sinx%5Ccdot+ln4-cosx%29%5C%5C%5C%5CI%281%2Bln%5E24%29%3D4%5E%7Bx%7D%28sinx%5Ccdot+ln4-cosx%29)
![I=\int 4^{x}\cdot sinx\, dx= \frac{sinx\cdot ln4-cosx}{1+ln^24} I=\int 4^{x}\cdot sinx\, dx= \frac{sinx\cdot ln4-cosx}{1+ln^24}](https://tex.z-dn.net/?f=I%3D%5Cint+4%5E%7Bx%7D%5Ccdot+sinx%5C%2C+dx%3D+%5Cfrac%7Bsinx%5Ccdot+ln4-cosx%7D%7B1%2Bln%5E24%7D+)
Artem990710:
Спасибо огромное
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