Question

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

• A. $e^{x}sinx+C$
• B. $e^{x}cosx+C$
• C. $e^x+cosx+C$
• D. $\text{None of these}$

Answer 1

The correct option is A

$e^{x}sinx+C$

Whenever you come across an integral which has $e^x$ with some other function in its integrand, it is advised to check whether the integral is of the form of $\int e^x\left(f\left(x\right)+f\text{'}\left(x\right)\right]dx$. In that case, we can use the formula $\int e^x\left(f\left(x\right)+f\text{'}\left(x\right)\right]dx=e^{x}f\left(x\right)$

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