Q93 / 180 · COMEDK 2014 · Mathematics

$\int e^{x}(\cos x-\sin x) d x$ is equal to

  • Correct answer
    A
    $e^{x} \cos x+C$
  • B
    $e^{x} \sin x+C$
  • C
    $-e^{x} \cos x+C$
  • D
    $-e^{x} \sin x+C$

Solution

$$ \begin{aligned} \text{Let} \quad I &=\int e^{x}(\cos x-\sin x) d x \\ &=\int e^{x} \cos x d x-\int e^{x} \sin x d x \\ &=\int e^{x} \cos x d x-\left[e^{x}(-\cos x)-\int e^{x}(-\cos x) d x\right] \\ &=\int e^{x} \cos x d x+e^{x} \cos x-\int e^{x} \cos x d x \\ &=e^{x} \cos x+C \end{aligned} $$

Hence, the correct answer is option: 'A'.

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