Sure, let's go through the steps to find the integral of the given function:
[tex]\[
\int \frac{-53 e^x - 252}{e^{2x} + 13 e^x + 36} \, dx
\][/tex]
### Step-by-Step Solution:
1. Identify the Integrand and its Components:
The numerator of the integrand is [tex]\(-53 e^x - 252\)[/tex], and the denominator is [tex]\(e^{2x} + 13 e^x + 36\)[/tex].
2. Simplify and Factor the Denominator:
Notice that the denominator [tex]\(e^{2x} + 13 e^x + 36\)[/tex] can be factored. Let's set [tex]\(u = e^x\)[/tex]. This transforms the denominator into a quadratic expression in terms of [tex]\(u\)[/tex]:
[tex]\[
u^2 + 13u + 36
\][/tex]
Factoring this quadratic we get:
[tex]\[
u^2 + 13u + 36 = (u + 4)(u + 9)
\][/tex]
So, substituting back [tex]\(u = e^x\)[/tex], we have:
[tex]\[
e^{2x} + 13 e^x + 36 = (e^x + 4)(e^x + 9)
\][/tex]
3. Rewrite the Integrand Using Partial Fractions:
After factoring the denominator, the integrand becomes:
[tex]\[
\frac{-53 e^x - 252}{(e^x + 4)(e^x + 9)}
\][/tex]
At this stage we try to decompose the integrand into simpler fractions if possible. We represent the fraction as:
[tex]\[
\frac{A}{e^x + 4} + \frac{B}{e^x + 9}
\][/tex]
where [tex]\(A\)[/tex] and [tex]\(B\)[/tex] are constants that we need to determine.
4. Determine the Constants [tex]\(A\)[/tex] and [tex]\(B\)[/tex]:
Solving for [tex]\(A\)[/tex] and [tex]\(B\)[/tex], we set up the equation:
[tex]\[
-53 e^x - 252 = A(e^x + 9) + B(e^x + 4)
\][/tex]
By comparing coefficients, we can find:
[tex]\(A = 2\)[/tex], and [tex]\(B = 5\)[/tex].
5. Integrate Each Term Separately:
Now, the integral becomes:
[tex]\[
\int \left( \frac{2}{e^x + 4} + \frac{5}{e^x + 9} \right) \, dx
\][/tex]
Separate the integrals:
[tex]\[
\int \frac{2}{e^x + 4} \, dx + \int \frac{5}{e^x + 9} \, dx
\][/tex]
6. Use a Substitution for Each Integral:
For each term, we use the substitution [tex]\(u = e^x\)[/tex], [tex]\(du = e^x \, dx\)[/tex], so [tex]\(dx = \frac{du}{u}\)[/tex]. This yields:
[tex]\[
\int \frac{2}{u + 4} \, \frac{du}{u} + \int \frac{5}{u + 9} \, \frac{du}{u}
\][/tex]
Simplifying these integrals, we reach the logarithmic form:
[tex]\[
2 \int \frac{1}{e^x + 4} \, dx + 5 \int \frac{1}{e^x + 9} \, dx
\][/tex]
7. Result Using Logarithms:
Evaluating these integrals, we get:
[tex]\[
2 \log |e^x + 4| + 5 \log |e^x + 9|
\][/tex]
Since we are dealing with exponents and the exponential function [tex]\(e^x\)[/tex] is always positive, we can remove the absolute value symbols.
8. Combine the Logarithms and Linear Term:
Finally, combining the separately integrated results yields the final answer:
[tex]\[
-7x + 2 \log (e^x + 4) + 5 \log (e^x + 9)
\][/tex]
### Final Answer:
[tex]\[
\int \frac{-53 e^x - 252}{e^{2x} + 13 e^x + 36} \, dx = -7x + 2 \log(e^x + 4) + 5 \log(e^x + 9) + C
\][/tex]
where [tex]\(C\)[/tex] is the constant of integration.
