Integrate the function in Exercise: e^x( + ) - Vidyadip

Question

Integrate the function in Exercise:
$\text{e}^\text{x}(\sin\text{x}+\cos\text{x})$

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Answer

Let $\text{I}=\int\text{e}^\text{x}(\sin\text{x}+\cos\text{x})\text{dx}$
Let $\text{f}(\text{x})=\sin\text{x}$
$\Rightarrow \ \text{f}'(\text{x})=\cos\text{x}$
$\therefore\ \text{I}=\int\text{e}^\text{x}\{\text{f}(\text{x})+\text{f}'(\text{x})\}\text{dx}$
It is known that, $=\int\text{e}^\text{x}\{\text{f}(\text{x})+\text{f}'(\text{x})\}\text{dx}=\text{e}^\text{x}\text{f}(\text{x})+\text{C}$
$\therefore\ \text{I}=\text{e}^\text{x}\sin\text{x}+\text{C}$

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