Question

Evaluate the following integrals. Include absolute values only when needed. $$\int \frac{e^{2 x}}{4+e^{2 x}} d x$$

Short answer

Short Answer: The solution to the given integral is $$\frac{1}{2} \ln|4 + e^{2x}| + C$$.

Step-by-step solution

Choose a substitution

Let us choose the substitution: $$u = 4 + e^{2x}$$

Differentiate the substitution

Differentiate both sides with respect to x: $$\frac{d u}{d x} = 0 + 2e^{2x}$$ Now, solve for \(dx\): $$d x = \frac{1}{2e^{2x}} d u$$

Substitute into the integral

Replace the terms with the substitution (\(4 + e^{2x}\) with \(u\) and \(dx\) with the expression found in step 2): $$\int \frac{e^{2 x}}{4+e^{2 x}} d x = \int \frac{e^{2x}}{u} \cdot \frac{1}{2e^{2x}} d u$$ The \(e^{2x}\) terms cancel out, and we are left with: $$\int \frac{1}{2u} d u$$

Solve the new integral

The new integral simplifies to: $$\frac{1}{2} \int \frac{1}{u} d u$$ This is a simple integral, and its solution is given by: $$\frac{1}{2} \ln |u| + C$$

Substitute back for u

Replace \(u\) with the original expression (\(4+e^{2x}\)): $$\frac{1}{2} \ln |4 + e^{2x}| + C$$ This is the solution to the given integral.