Which of the following is equivalent to $\int_{3}^{5} x \ln x d x$ ?
(A) $\quad \int_{\ln 3}^{\ln 5} u d u$
(B) $\quad \int_{3}^{5} x d x \cdot \int_{3}^{5} \ln x d x$
(C) $\left.\quad \frac{1}{2} x^{2} \ln x\right|_{3} ^{5}-\int_{3}^{5} \frac{1}{2} x d x$
(D) $\left.\quad \frac{1}{2} x^{2} \ln x\right|_{3} ^{5}+\int_{3}^{5} \frac{1}{2} x d x$

Step 1 ：The integral \(\int_{3}^{5} x \ln x d x\) is a product of two functions, \(x\) and \(\ln x\). This is a perfect candidate for integration by parts, which is a method of integration that is used when the integral is a product of two functions.

Step 2 ：The formula for integration by parts is \(\int u dv = uv - \int v du\). We can let \(u = \ln x\) and \(dv = x dx\).

Step 3 ：Then, we need to find \(du\) and \(v\). We know that \(du = \frac{1}{x} dx\) and \(v = \frac{1}{2} x^2\).

Step 4 ：Plugging these into the integration by parts formula, we get \(\int_{3}^{5} x \ln x d x = \left.\frac{1}{2} x^{2} \ln x\right|_{3} ^{5} - \int_{3}^{5} \frac{1}{2} x d x\).

Step 5 ：This matches with option (C). So, the equivalent form of the given integral is option (C).

Step 6 ：Final Answer: The equivalent form of the given integral is \(\boxed{\text{(C)} \left.\frac{1}{2} x^{2} \ln x\right|_{3} ^{5} - \int_{3}^{5} \frac{1}{2} x d x}\).