Find the integral. Select the correct answer.
$$\int x ^ { 5 } \ln x d x$$
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

We use integration by parts:

$$u= \ln x, dv=x^5dx $$

$$du=\frac{1}{x}dx, v=\frac{1}{6}x^6$$

Using the formula:

$$\int u dv= uv- \int v du$$

We get:

$$ \int x ^ { 5 } \ln x d x= \frac { 1 } { 6 } x ^ { 6 } \ln x - \int \frac { 1 } { 6 } x ^ { 6 } \cdot \frac { 1 } { x } d x = \frac { 1 } { 6 } x ^ { 6 } \ln x - \frac { 1 } { 36 } x ^ { 6 } + C$$

Simplifying, we have:

$$\int x ^ { 5 } \ln x d x= \frac { 1 } { 36 } x ^ { 6 } ( 6 \ln x - 1 ) + C$$

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

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