Question

$\int \left\{\sin \right(\ln x)+\cos (\ln x\left)\right\}dx$ is equal to

• A. $\ln x\cdot \sin \left(\ln x\right)+C$
• B. $x\cdot \cos \left(e^x\right)+C$
• C. $x\cdot \sin \left(\ln x\right)+C$
• D. $x\cdot \cos \left(\ln x\right)+C$

Answer 1

The correct option is C $x\cdot \sin \left(\ln x\right)+C$

$I=\int \left\{\sin \right(\ln x)+\cos (\ln x\left)\right\}dx$
Putting $\ln x=t$
$\Rightarrow x=e^t$
$\Rightarrow dx=e^{t}dt$
$\Rightarrow I=e^t(\sin t+\cos t)dt\underset{f\left(x\right)}{\downarrow }\underset{f^{\prime }\left(x\right)}{\downarrow }\{\because \int e^x\left\{f\right(x)+f^{\prime }(x\left)\right\}dx=e^{x}f\left(x\right)+C\}$
$\Rightarrow I=e^t\cdot \sin t+C=x\cdot \sin \left(\ln x\right)+C$