MCQ
$\int \mathrm{e}^{\mathrm{x}}\left\{\mathrm{f}(\mathrm{x})+\mathrm{f}^{\prime}(\mathrm{x})\right\} \mathrm{dx}=$
- A$2 e^{x} f(x)+C$
- B$e^{x}-f(x)+C$
- C$e^{x} f(x)+C$
- D$e^{x}+f(x)+C$
✓
Answer
(C) $e^{x} f(x)+C$
Explanation: $\int \mathrm{e}^{\mathrm{x}}\left\{\mathrm{f}(\mathrm{x})+\mathrm{f}^{\prime}(\mathrm{x})\right\} \mathrm{dx}=\mathrm{e}^{\mathrm{x}} \mathrm{f}(\mathrm{x})+\mathrm{C}$
$\mathrm{t}=\mathrm{e}^{\mathrm{x}} \mathrm{f}(\mathrm{x})$
$\frac{d t}{d x}=e^{x} \cdot \frac{d}{d x}(f(x))+f(x) \frac{d}{d x}\left(e^{x}\right)$
$=\mathrm{e}^{\mathrm{x}} \mathrm{f}^{\prime}(\mathrm{x})+\mathrm{f}(\mathrm{x}) \cdot \mathrm{e}^{\mathrm{x}}$
$d t=e^{x}\left(f^{\prime}(x)+f(x)\right) d x$
$\int \mathrm{e}^{\mathrm{x}}\left\{\mathrm{f}(\mathrm{x})+\mathrm{f}^{\prime}(\mathrm{x})\right\} \mathrm{dx}=\int \mathrm{dt}=\mathrm{t}+\mathrm{C}$
$=\mathrm{e}^{\mathrm{x}} \mathrm{f}(\mathrm{x})+\mathrm{C}$
Explanation: $\int \mathrm{e}^{\mathrm{x}}\left\{\mathrm{f}(\mathrm{x})+\mathrm{f}^{\prime}(\mathrm{x})\right\} \mathrm{dx}=\mathrm{e}^{\mathrm{x}} \mathrm{f}(\mathrm{x})+\mathrm{C}$
$\mathrm{t}=\mathrm{e}^{\mathrm{x}} \mathrm{f}(\mathrm{x})$
$\frac{d t}{d x}=e^{x} \cdot \frac{d}{d x}(f(x))+f(x) \frac{d}{d x}\left(e^{x}\right)$
$=\mathrm{e}^{\mathrm{x}} \mathrm{f}^{\prime}(\mathrm{x})+\mathrm{f}(\mathrm{x}) \cdot \mathrm{e}^{\mathrm{x}}$
$d t=e^{x}\left(f^{\prime}(x)+f(x)\right) d x$
$\int \mathrm{e}^{\mathrm{x}}\left\{\mathrm{f}(\mathrm{x})+\mathrm{f}^{\prime}(\mathrm{x})\right\} \mathrm{dx}=\int \mathrm{dt}=\mathrm{t}+\mathrm{C}$
$=\mathrm{e}^{\mathrm{x}} \mathrm{f}(\mathrm{x})+\mathrm{C}$
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