  A study of $B_c^+ \rightarrow \chi_c \pi^+$ decays is reported using
proton-proton collision data, collected with the LHCb detector at
centre-of-mass energies of 7, 8, and 13 TeV, corresponding to an integrated
luminosity of 9fb$^{-1}$. The decay $B_c^+ \rightarrow \chi_{c2} \pi^+$ is
observed for the first time, with a significance exceeding seven standard
deviations. The relative branching fraction with respect to the $B_c^+
\rightarrow J/\psi \pi^+$ decay is measured to be $$ \frac{\mathcal{B}_{B_c^+
\rightarrow \chi_{c2} \pi^+}}
  {\mathcal{B}_{B_c^+ \rightarrow J/\psi \pi^+}} =
  0.37 \pm 0.06 \pm 0.02 \pm 0.01 , $$ where the first uncertainty is
statistical, the second is systematic, and the third is due to the knowledge of
the $\chi_c \rightarrow J/\psi \gamma$ branching fraction. No significant
$B_c^+ \rightarrow \chi_{c1} \pi^+$ signal is observed and an upper limit for
the relative branching fraction for the $B_c^+ \rightarrow \chi_{c1} \pi^+$ and
$B_c^+ \rightarrow \chi_{c2} \pi^+$ decays of $$ \frac{\mathcal{B}_{B_c^+
\rightarrow \chi_{c1} \pi^+}}
  {\mathcal{B}_{B_c^+ \rightarrow \chi_{c2} \pi^+}} < 0.49 $$ is set at the
90\% confidence level.
