This can be solved in a far simpler way: -
So, to get the RMS value of the waveform, you square the individual parts (triangle and DC) to get power into a 1 Ω resistor, weight them with their duty cycle, add them together and finally, take the square root to get back to RMS voltage and lose the 1 Ω dependency.
- For the triangle section, it's weighted power is \$\dfrac{4}{3}\times 7\div 15\$
- For the DC part it's just \$4\times 8\div 15\$
- Add them to get 2.755555
- Take the square root to get 1.65998661307
Proof of triangle waveform RMS: -
