Integral of \( \displaystyle \frac{e^{2 x - 1}}{e^{2 x - 1} + 1} \)

Problem 4.658 · medium

1. \[ \int \frac{e^{2 x - 1}}{e^{2 x - 1} + 1}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \frac{e^{2 x}}{e \left(\frac{e^{2 x}}{e} + 1\right)}\, dx \]
   algebraRewrite the exponent using exponent laws.≈ Checked numerically
3. \[ = \int \frac{e^{2 x}}{e^{2 x} + e}\, dx \]
   algebra algebraSimplify the fraction by multiplying numerator and denominator by exp(1). Rewrite the integrand to reveal the derivative of the denominator.✓ Proved
4. \[ = \frac{\ln{\left(e^{2 x} + e \right)}}{2} \]
   antiderivativeIntegrate using the substitution method or logarithmic rule.✓ Proved
5. \[ = \frac{\ln{\left(e e^{2 x - 1} + e \right)}}{2} \]
   algebraRewrite the first term inside the log to match the original form.✓ Proved
6. \[ = \frac{\ln{\left(e \left(e^{2 x - 1} + 1\right) \right)}}{2} \]
   algebraFactor out exp(1) from the expression inside the logarithm.✓ Proved
7. \[ = \frac{\ln{\left(e^{2 x - 1} + 1 \right)}}{2} \]
   simplifyUse log properties to simplify the expression.≈ Checked numerically