To evaluate the integral [tex]\(\int_1^e \left(16x - \frac{5}{x}\right) \, dx\)[/tex], follow these steps:
1. Separate the integral into two parts:
[tex]\[
\int_1^e \left(16x - \frac{5}{x}\right) \, dx = \int_1^e 16x \, dx - \int_1^e \frac{5}{x} \, dx
\][/tex]
2. Compute each integral individually:
- For the first integral [tex]\(\int_1^e 16x \, dx\)[/tex]:
[tex]\[
\int 16x \, dx = 16 \int x \, dx = 16 \left(\frac{x^2}{2}\right) = 8x^2
\][/tex]
Evaluate this result from 1 to [tex]\(e\)[/tex]:
[tex]\[
\left. 8x^2 \right|_1^e = 8e^2 - 8
\][/tex]
- For the second integral [tex]\(\int_1^e \frac{5}{x} \, dx\)[/tex]:
[tex]\[
\int \frac{5}{x} \, dx = 5 \int \frac{1}{x} \, dx = 5 \ln|x|
\][/tex]
Evaluate this result from 1 to [tex]\(e\)[/tex]:
[tex]\[
\left. 5 \ln|x| \right|_1^e = 5 \ln e - 5 \ln 1 = 5 \cdot 1 - 5 \cdot 0 = 5
\][/tex]
3. Combine the evaluated results:
[tex]\[
\int_1^e \left(16x - \frac{5}{x}\right) \, dx = (8e^2 - 8) - 5
\][/tex]
Simplify the expression:
[tex]\[
8e^2 - 8 - 5 = 8e^2 - 13
\][/tex]
So, the value of the integral is [tex]\(\boxed{8e^2 - 13}\)[/tex].
Thus, the correct choice is:
A. [tex]\(8e^2 - 13\)[/tex]
