Find the integral.
$$\int x ^ { 5 } \ln x d x$$
A) $\frac { 1 } { 6 } x ^ { 6 } ( 6 \ln x - 1 ) + C$
B) $\frac { 1 } { 36 } x ^ { 6 } ( \ln x - 1 ) + C$
C) $\frac { 1 } { 6 } x ^ { 5 } + \frac { 1 } { x } + C$
D) $\frac { 1 } { 36 } x ^ { 6 } ( 6 \ln x - 1 ) + C$

Question

Quizplus · Accepted Answer

This problem requires the use of integration by parts, with $u=\ln x$ and $dv=x^5dx$. Then $du=\frac{1}{x}dx$ and $v=\frac{1}{6}x^6$. Thus, we have \begin{align*}
\int x^5\ln xdx &= \frac{1}{6}x^6\ln x-\int \frac{1}{6}x^6\cdot\frac{1}{x}dx\
&=\frac{1}{6}x^6\ln x - \frac{1}{36}x^6 + C\
&=\frac{1}{36}x^6(6\ln x-1) + C.
\end{align*}Therefore, the answer is $\boxed{	extbf{(D)}}$.

Find the Integral ∫x5ln⁡xdx\int x ^ { 5 } \ln x d x∫x5lnxdx

Find the Integral $\int x ^ { 5 } \ln x d x$