Evaluate the integral by using a substitution prior to integration by parts.
-$$\int \cos ( \ln x ) d x$$
A) $x [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$
B) $\frac { x } { 2 } [ \cos ( \ln x ) + \sin ( \ln x ) ] + C$
C) $x \cos ( \ln x ) + \sin ( \ln x ) + C$
D) $\frac { x } { 2 } [ \cos ( \ln x ) - \sin ( \ln x ) ] + C$

Question

Quizplus · Accepted Answer

Using the substitution $ u = \ln x $, $ du = \frac{1}{x} dx $, the integral becomes $\int \cos(u) e^u du$. Applying integration by parts with $ v = e^u $ and $ dw = \cos(u) du $, we get $\frac{x}{2}[\cos(\ln x) + \sin(\ln x)] + C$.

Evaluate the Integral by Using a Substitution Prior to Integration ∫cos⁡(ln⁡x)dx\int \cos ( \ln x ) d x∫cos(lnx)dx

Evaluate the Integral by Using a Substitution Prior to Integration $\int \cos ( \ln x ) d x$