To solve the integral [tex]\(\int (2 \tan x - 5 e^x) \, dx\)[/tex], we can break the integral into parts and solve each part separately:
1. First, let's integrate [tex]\(2 \tan x\)[/tex]:
[tex]\[
\int 2 \tan x \, dx
\][/tex]
We know that the integral of [tex]\(\tan x\)[/tex] is [tex]\(\log|\sec x|\)[/tex], but using another trigonometric identity, [tex]\(\sec x = \frac{1}{\cos x}\)[/tex], we get:
[tex]\[
\int 2 \tan x \, dx = 2 \int \tan x \, dx = 2 \log|\sec x| = 2 \log \left| \frac{1}{\cos x} \right|
\][/tex]
Since [tex]\(|\sec x| = \frac{1}{|\cos x|}\)[/tex], and for simplicity, assuming [tex]\( \cos x \)[/tex] is positive, we can write:
[tex]\[
2 \log \left| \frac{1}{\cos x} \right| = -2 \log|\cos x|
\][/tex]
2. Next, let's integrate [tex]\(-5 e^x\)[/tex]:
[tex]\[
\int -5 e^x \, dx
\][/tex]
We know that the integral of [tex]\(e^x\)[/tex] is [tex]\(e^x\)[/tex]. Therefore:
[tex]\[
\int -5 e^x \, dx = -5 \int e^x \, dx = -5 e^x
\][/tex]
3. Combining both parts, we get the integral of the original function:
[tex]\[
\int (2 \tan x - 5 e^x) \, dx = -5 e^x - 2 \log|\cos x| + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
Thus, the solution to the integral [tex]\(\int (2 \tan x - 5 e^x) \, dx\)[/tex] is:
[tex]\[
-5 e^x - 2 \log|\cos x| + C
\][/tex]
