Evaluate \[ \int e^x\cos x\,dx \qquad\text{and}\qquad \int\sin(\ln x)\,dx, \quad x>0. \]
Solution
Integrating the first integral by parts twice yields \[ \boxed{\int e^x\cos x\,dx=\frac12e^x(\cos x+\sin x)+C.} \] For the second, let \(u=\ln x\), so \(dx=e^u du\). Then \[ \int\sin(\ln x)\,dx=\int e^u\sin u\,du =\boxed{\frac x2\left(\sin(\ln x)-\cos(\ln x)\right)+C.} \]