To tackle the integral
[tex]\[
\int \frac{e^{2x}}{49 + e^{4x}} \, dx,
\][/tex]
we need to employ a strategy that simplifies the integrand. Here is a detailed, step-by-step breakdown of solving this integral.
### Step 1: Rewrite the Integrand
First, observe that the denominator [tex]\(49 + e^{4x}\)[/tex] can present complications. However, let's rewrite the integrand in a more insightful way:
[tex]\[
\frac{e^{2x}}{49 + e^{4x}} = \frac{e^{2x}}{49 + (e^{2x})^2}.
\][/tex]
### Step 2: Set a Substitution
Let [tex]\(u = e^{2x}\)[/tex]. Then, we differentiate:
[tex]\[
\frac{du}{dx} = 2e^{2x} \implies du = 2e^{2x} \, dx \implies e^{2x} \, dx = \frac{1}{2} \, du.
\][/tex]
Thus, the integral becomes:
[tex]\[
\int \frac{e^{2x}}{49 + u^2} \cdot \frac{1}{2} \, du = \frac{1}{2} \int \frac{1}{49 + u^2} \, du.
\][/tex]
### Step 3: Apply the Standard Integral Formula
The integral we now need to solve is:
[tex]\[
\frac{1}{2} \int \frac{1}{49 + u^2} \, du.
\][/tex]
We recognize this as a standard form:
[tex]\[
\int \frac{1}{a^2 + u^2} \, du = \frac{1}{a} \arctan \left( \frac{u}{a} \right) + C.
\][/tex]
In our case, [tex]\(a^2 = 49\)[/tex], so [tex]\(a = 7\)[/tex]. Therefore, the integral transforms to:
[tex]\[
\frac{1}{2} \cdot \frac{1}{7} \arctan \left( \frac{u}{7} \right) + C = \frac{1}{14} \arctan \left( \frac{u}{7} \right) + C.
\][/tex]
### Step 4: Substitute Back the Original Variable
Recall that [tex]\(u = e^{2x}\)[/tex]. Substitute back to obtain:
[tex]\[
\frac{1}{14} \arctan \left( \frac{e^{2x}}{7} \right) + C.
\][/tex]
### Final Answer
After all steps, the integral we sought is:
[tex]\[
\int \frac{e^{2x}}{49 + e^{4x}} \, dx = \frac{1}{14} \arctan \left( \frac{e^{2x}}{7} \right) + C.
\][/tex]
This would be the step-by-step way of solving this particular integral.
