Evaluate the integral $\int (e^{\ln x} + \sin x) \cos x dx$. Which of the following expressions is/a

Question

Evaluate the integral $\int (e^{\ln x} + \sin x) \cos x dx$. Which of the following expressions is/are correct? (Assume $x>0$ and $C$ is a constant of integration)

Professor PC Thomas Classes · Answer

Since $e^{\ln x} = x$, $I = \int x\cos x dx + \int \sin x \cos x dx$. The first part by parts is $x\sin x - \int \sin x dx = x\sin x + \cos x$. The second part can be written as $\frac{\sin^2 x}{2}$ or $-\frac{\cos 2x}{4}$. Adding them gives Option A and Option C (they differ only by a constant $1/4$, which is absorbed by $C$).