Question

If an antiderivative of $f(x)$ is $e^x$ and that of $g(x)$ is $\cos x$, then $\int f(x) \cos x d x+\int g(x) e^x d x$ is equal to
where $C$ is constant of integration.

• A. $f(x) g(x)+C$
• B. $f(x)+g(x)+C$
• C. $ e ^{ x } \cos x + C$
• D. $f(x)-g(x)+C$

Answer 1

Answer: (C)

$ \because \int f(x) d x=e^x \Rightarrow f(x)=e^x$
$\int g(x) d x=\cos x \Rightarrow g(x)=-\sin x$
$\text { Given expression } =\int e^x \cos x d x+\int-\sin x e^x d x $
$ =\int e^x(\cos x+(-\sin x)) d x=e^x \cos x+C$