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

Problem 4.996 · medium

1. \[ \int \frac{e^{2 x + 1}}{e^{2 x + 1} + 1}\, dx \]
   integral algebraStart with the integral of the given function. Rewrite the numerator to facilitate substitution.✓ Proved
2. \[ = \int \frac{2 e^{2 x + 1}}{2 e^{2 x + 1} + 2}\, dx \]
   algebraMultiply and divide by 2.✓ Proved
3. \[ = \int \frac{e^{2 x + 1}}{e^{2 x + 1} + 1}\, dx \]
   simplifySimplify the expression.✓ Proved
4. \[ = \int \frac{1}{e^{- 2 x - 1} + 1}\, dx \]
   algebra algebraDivide numerator and denominator by exp(2*x + 1). Rewrite 1/exp(2*x + 1) as exp(-2*x - 1).✓ Proved
5. \[ = \int \frac{e^{2 x + 1}}{e^{2 x + 1} + 1}\, dx \]
   simplify algebra algebraReturn to the original form for easier integration. Prepare for substitution by identifying the derivative of the denominator. Isolate the derivative of the denominator.✓ Proved
6. \[ = \frac{\ln{\left(e^{2 x + 1} + 1 \right)}}{2} \]
   antiderivativeIntegrate using the substitution u = exp(2*x + 1) + 1, du = 2*exp(2*x + 1) dx.✓ Proved