$\text { Find } \int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x$
A)$\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \frac { 1 } { 12 } \ln \left| \ln \left( x ^ { 12 } \right) \right| + C$
B)$\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \ln \left| x ^ { 12 } \right| + C$
C) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = 12 \ln \left| \ln \left( x ^ { 12 } \right) \right| + C$
D)$\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = 12 \ln \left| x ^ { 12 } \right| + C$
E)$\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \frac { 1 } { 12 } \ln \left| x ^ { 12 } \right| + C$

Question

$\text { Find } \int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x$ 
A)$\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \frac { 1 } { 12 } \ln \left| \ln \left( x ^ { 12 } \right) \right| + C$
B)$\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \ln \left| x ^ { 12 } \right| + C$
C) $\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = 12 \ln \left| \ln \left( x ^ { 12 } \right) \right| + C$
D)$\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = 12 \ln \left| x ^ { 12 } \right| + C$
E)$\int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x = \frac { 1 } { 12 } \ln \left| x ^ { 12 } \right| + C$

Quizplus · Accepted Answer

We can start by using substitution, $u = \ln \left( x^{12} \right) = 12 \ln(x)$. This gives us $du = \frac{12}{x}dx$. 
Substituting back, we get $\int \frac{1}{x \ln \left( x^{12} \right)} dx = \frac{1}{12} \int \frac{1}{u} du = \frac{1}{12} \ln \left| u \right| + C$. 
Substituting back again, we get $\frac{1}{12} \ln \left| \ln \left( x^{12} \right) \right| + C$. Therefore, the correct answer is (A).

Find ∫1xln⁡(x12)dx\text { Find } \int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x Find ∫xln(x12)1​dx

$\text { Find } \int \frac { 1 } { x \ln \left( x ^ { 12 } \right) } d x$