The integral (_ex),dx is equal to : (where C is a constant of integration)

Question

The integral $\displaystyle\int\cos(\log_ex)\,dx$ is equal to : (where C is a constant of integration)

Accepted Answer

Let $I=\displaystyle\int\cos(\ln x)\,dx$. Integrating by parts:
$$I=x\cos(\ln x)+\int\sin(\ln x)\,dx=x\cos(\ln x)+\left[x\sin(\ln x)-\int\cos(\ln x)\,dx\right]$$

$$\Rightarrow2I=x[\cos(\ln x)+\sin(\ln x)]+C\Rightarrow I=\frac x2[\cos(\ln x)+\sin(\ln x)]+C$$

Solutions · 1