Separate: \[\int\frac5{x+5}\,dx=\int\sin^22\theta\,d\theta.\] Using \(\sin^22\theta=(1-\cos4\theta)/2\), \[5\ln|x+5|=\frac\theta2-\frac18\sin4\theta+C.\] At \(\theta=0,x=5\), \(C=5\ln10\). On this branch \(x+5>0\); divide by 5 and exponentiate: \[x=10\exp\left(\frac\theta{10}-\frac1{40}\sin4\theta\right)-5.\]