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

Question

Quizplus · Accepted Answer

To solve the integral $\int x \ln \sqrt { x } d x$, we use integration by parts, where $u = \ln \sqrt{x} = \frac{1}{2} \ln x$ and $dv = x dx$. Differentiating $u$ gives $du = \frac{1}{2x} dx$, and integrating $dv$ gives $v = \frac{x^2}{2}$. Applying the integration by parts formula, $\int u dv = uv - \int v du$, we get $\frac{x^2}{2} \cdot \frac{1}{2} \ln x - \int \frac{x^2}{2} \cdot \frac{1}{2x} dx = \frac{x^2 \ln x}{4} - \int \frac{x}{4} dx = \frac{x^2 \ln x}{4} - \frac{x^2}{8} + C$, which matches option C.

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

Evaluate $\int x \ln \sqrt { x } d x$
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