Integral of \( \displaystyle \tan^{2}{\left(3 x \right)} \)

Problem 4.281 · easy

1. \[ \int \tan^{2}{\left(3 x \right)}\, dx \]
   integralStart with the integral of the given function.✓ Proved
2. \[ = \int \left(-1 + \frac{1}{\cos^{2}{\left(3 x \right)}}\right)\, dx \]
   trig-identityUse the identity tan(z)^2 = sec(z)^2 - 1, where sec(z) = 1/cos(z).✓ Proved
3. \[ = - \int 1\, dx + \int \sec^{2}{\left(3 x \right)}\, dx \]
   linearitySplit the integral into two parts.✓ Proved
4. \[ = - x + \int \sec^{2}{\left(3 x \right)}\, dx \]
   antiderivativeThe antiderivative of 1 is x.✓ Proved
5. \[ = - x + \frac{\tan{\left(3 x \right)}}{3} \]
   antiderivativeThe antiderivative of sec(3*x)^2 is tan(3*x)/3.✓ Proved