Question

If the integral of the function $\frac{\sin \left(\ln x\right)}{x}$ is f(x), then the value of f(1) is if the constant of integration is zero.

• A. 0.0
• B. 1
• C. -1
• D. 1/2

Answer 1

The correct option is C -1

We know the integral of $\sin x$ and $\ln x$ separately but not of the function $\sin \left(\ln x\right)$. But we have the derivative of $\ln x$ in the given function. We will replace $lnx$ with t and proceed
$t=\ln x$
$\Rightarrow dt=\frac{dx}{x}$
$\Rightarrow \int \frac{\sin \left(\ln x\right)}{x}dx=\int sin\left(t\right)dt=-\cos t+c$
Replacing t with $\ln x$, we get
$f\left(x\right)=-\cos \left(\ln x\right)$
We want to find f(1)
$f\left(1\right)=-\cos \left(ln\right(1\left)\right)$
$=-\cos \left(0\right)$
= -1