Evaluate the integral.
-$\int _ { 1 } ^ { 2 } x \ln x d x$
A) $\frac { 3 } { 4 }$ + ln 2
B) 2 ln 2 - 3/4
C) 2 ln 2
D) ln 2 - $\frac { 1 } { 2 }$

Question

Evaluate the integral. 
-$\int _ { 1 } ^ { 2 } x \ln x d x$
A) $\frac { 3 } { 4 }$ + ln 2
B) 2 ln 2 - 3/4
C) 2 ln 2
D) ln 2 - $\frac { 1 } { 2 }$

Quizplus · Accepted Answer

To solve the integral $\int _ { 1 } ^ { 2 } x \ln x d x$, we use integration by parts, where $u = \ln x$ and $dv = x dx$. This gives us $du = \frac{1}{x} dx$ and $v = \frac{1}{2}x^2$. Applying the formula for integration by parts, $\int u dv = uv - \int v du$, we get $\frac{1}{2}x^2 \ln x - \int \frac{1}{2}x^2 \cdot \frac{1}{x} dx = \frac{1}{2}x^2 \ln x - \frac{1}{4}x^2$, evaluated from 1 to 2. This simplifies to $2 \ln 2 - \frac{3}{4}$.

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Evaluate the Integral $\int _ { 1 } ^ { 2 } x \ln x d x$