Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral.
I. Integration by parts
II.
III.
IV.

A) I. Integration by parts;
D)
E) not integrable by any of the techniques we have studied

Question

Determine the most appropriate method or integral formula for evaluating the given integral. Next, evaluate the integral. ​
I. Integration by parts 
II. 
III. 
IV.

A) I. Integration by parts; 
D) ​ 
E) not integrable by any of the techniques we have studied

Quizplus · Accepted Answer

The given integral can be evaluated using integration by parts with $u = \ln(x)$ and $dv = x^3 \, dx$. Thus, $du = \frac{1}{x} \, dx$ and $v = \frac{x^4}{4}$. Using the integration by parts formula, we have:

$$\int \ln(x) \cdot x^3 \, dx = \frac{x^4}{4} \ln(x) - \int \frac{x^3}{4} \, dx = \frac{x^4}{4} \ln(x) - \frac{x^4}{16} + C$$

where $C$ is the constant of integration. Therefore, the correct choice is A.

Determine the Most Appropriate Method or Integral Formula for Evaluating