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Question Number 216799 by OPAVdx last updated on 20/Feb/25

Commented by MathematicalUser2357 last updated on 22/Feb/25

$$sen?$$

Answered by MATHEMATICSAM last updated on 22/Feb/25

$$\int e^x\sin xdx$$$$u=\sin x$$$$\Rightarrow du=\cos xdx$$$$dv=e^{x}dx$$$$\Rightarrow v=e^x$$$$$$$$\int udv=uv-\int vdu$$$$=e^x\sin x-\int e^x\cos xdx....\left(\mathrm{i}\right)$$$$$$$$\mathrm{Now}\mathrm{lets}\mathrm{integrate}\int e^x\cos xdx$$$$\mathrm{Now}u=\cos x$$$$\Rightarrow du=-\sin xdx$$$$dv=e^{x}dx$$$$\Rightarrow v=e^x$$$$\int udv=uv-\int vdu$$$$=e^x\cos x+\int e^x\sin xdx...\left(\mathrm{ii}\right)$$$$$$$$\mathrm{From}\left(\mathrm{i}\right)\mathrm{and}\left(\mathrm{ii}\right)$$$$\int e^x\sin xdx=e^x\sin x-e^x\cos x$$$$-\int e^x\sin xdx$$$$\Rightarrow 2\int e^x\sin xdx=e^x(\sin x-\cos x)$$$$\Rightarrow \int e^x\sin xdx=\frac{e^x(\sin x-\cos x)}{2}+\mathrm{C}$$

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