Integral of \( \displaystyle \tan^{2}{\left(2 x + 1 \right)} \)

Problem 4.920 · medium

1. \[ \int \tan^{2}{\left(2 x + 1 \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(\sec^{2}{\left(2 x + 1 \right)} - 1\right)\, dx \]
   trig-identityUse the identity tan(u)**2 = sec(u)**2 - 1.Reviewed
3. \[ = - \int 1\, dx + \int \sec^{2}{\left(2 x + 1 \right)}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
4. \[ = - x + \int \sec^{2}{\left(2 x + 1 \right)}\, dx \]
   antiderivativeIntegrate the constant term.✓ Proved
5. \[ = - x + \frac{\tan{\left(2 x + 1 \right)}}{2} \]
   antiderivativeIntegrate the secant squared term using substitution or known formula.Reviewed