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Question Number 136935 by metamorfose last updated on 27/Mar/21

$$\int cos\left(x\right)e^{cos\left(x\right)+sin\left(x\right)}dx=...?$$

Answered by liberty last updated on 27/Mar/21

$$\int \cos \mathrm{x}.{\mathrm{e}}^{\cos \mathrm{x}}.{\mathrm{e}}^{\sin \mathrm{x}}\mathrm{dx}=$$$$\mathrm{by}\mathrm{parts}\{\begin{array}{l}\mathrm{u}={\mathrm{e}}^{\cos \mathrm{x}}\mathrm{du}=-\sin \mathrm{x}.{\mathrm{e}}^{\cos \mathrm{x}}\mathrm{dx}\\ \mathrm{v}=\int \cos \mathrm{x}.{\mathrm{e}}^{\sin \mathrm{x}}\mathrm{dx}=\int \mathrm{d}\left({\mathrm{e}}^{\sin \mathrm{x}}\right)={\mathrm{e}}^{\sin \mathrm{x}}\end{array}$$$$\int {\mathrm{e}}^{\cos \mathrm{x}}\mathrm{d}\left({\mathrm{e}}^{\sin \mathrm{x}}\right)={\mathrm{e}}^{\cos \mathrm{x}+\sin \mathrm{x}}+\int \sin {\mathrm{xe}}^{\cos \mathrm{x}+\sin \mathrm{x}}\mathrm{dx}$$$$(1-\sin \mathrm{x})\int \cos \mathrm{x}.{\mathrm{e}}^{\cos \mathrm{x}+\sin \mathrm{x}}\mathrm{dx}={\mathrm{e}}^{\cos \mathrm{x}+\sin \mathrm{x}}$$$$\int \cos \mathrm{x}.{\mathrm{e}}^{\cos \mathrm{x}+\sin \mathrm{x}}\mathrm{dx}=\frac{{\mathrm{e}}^{\cos \mathrm{x}+\sin \mathrm{x}}+\mathrm{C}}{1-\sin \mathrm{x}}$$

Commented by metamorfose last updated on 28/Mar/21

$$Sir,i{can}^{\prime }tunderstandtheligne4$$

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