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 integral requires 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
$$\int x^5\ln xdx=\frac{1}{6}x^6\ln x-\int \frac{1}{x}\cdot\frac{1}{6}x^6dx=\frac{1}{6}x^6\ln x-\frac{1}{36}x^6+C.$$
Therefore, the correct answer is D.

Find the Integral ∫x5ln⁡xdx\int x^{5} \ln x d x∫x5lnxdx A) 16x6(6ln⁡x−1)+C\frac{1}{6} x^{6}(6 \ln x-1)+C61​x6(6lnx−1)+C

Find the Integral $\int x^{5} \ln x d x$ A) $\frac{1}{6} x^{6}(6 \ln x-1)+C$