Question

$\int e^x(\cos x-\sin x)dx$is equal to
(a) e^x cos x + C (b) e^x sin x + C
(c) -e^x cos x + C (d) -e^x sin x + C

Answer 1

$I=\int e^x\left(\cos x-\sin x\right)dx$

$\Rightarrow I=\int e^x\left[\cos x+\left(-\sin x\right)\right]dx$

Let $f\left(x\right)=\cos x$

$\Rightarrow f\text{'}\left(x\right)=-\sin x$

$\therefore I=e^x\cos x+C\left[\int e^x\left[f\left(x\right)+f\text{'}\left(x\right)\right]dx=e^{x}f\left(x\right)+C\right]$

Hence, the correct answer is option (a).