Question

$\int e^x(\sin x+\cos x)dx$ is equal to

• A. $e^x\sin x+C$
• B. $e^x\cos x+C$
• C. $e^x+\cos x+C$

Answer 1

The correct option is A $e^x\sin x+C$

We know that
$\int e^x\left[f\right(x)+f^{\prime }(x\left)\right]dx=e^{x}f\left(x\right).$

Here, we can see that cos x is the derivative of sin x, so the given integral is of the form $\int e^x\left[f\right(x)+f^{\prime }(x\left)\right]dx$, where f(x) is sin x.
Hence we can directly say that the solution of this would be $e^x\sin x$ + C.