Evaluate $\int x \ln \sqrt { x } d x$
A) x + C
B) $x ^ { 2 } \ln x + C$
C) $\frac { x ^ { 2 } \ln x } { 4 } + C$
D) $\frac { x ^ { 2 } \ln x } { 4 } - \frac { x ^ { 2 } } { 8 } + C$

Question

Quizplus · Accepted Answer

To solve $\int x \ln \sqrt { x } d x$, we first simplify the logarithm: $\ln \sqrt{x} = \frac{1}{2} \ln x$. The integral becomes $\int x \cdot \frac{1}{2} \ln x dx = \frac{1}{2} \int x \ln x dx$. Using integration by parts, where $u = \ln x$ and $dv = x dx$, we find $du = \frac{1}{x} dx$ and $v = \frac{x^2}{2}$. Thus, the integral becomes $\frac{1}{2} \left( \frac{x^2}{2} \ln x - \int \frac{x^2}{2} \cdot \frac{1}{x} dx \right) = \frac{x^2 \ln x}{4} - \frac{1}{2} \cdot \frac{x^2}{4} + C$, which simplifies to $\frac{x^2 \ln x}{4} - \frac{x^2}{8} + C$.

Evaluate ∫xln⁡xdx\int x \ln \sqrt { x } d x∫xlnx​dx A) X + C B)

Evaluate $\int x \ln \sqrt { x } d x$ A) X + C
B)