Six identities for sixth-order mock theta functions are proved. One of these was given by Ramanujan in the ``Lost Notebook'' and was subsequently studied by \textit{G. Andrews} and \textit{D. Hickerson} [Adv. Math. 89, No. 1, 60-105 (1991; Zbl 0739.11042)]. The other five identities are new. Ramanujan actually defined seven sixth-order mock theta functions. The results presented here involve three of them: \(\rho,\psi\) and \(\lambda\). There are two results given for each function. The first three results are \(q\)-series identities. For example, one of these results is \[ \psi (q^2)+2 \varphi(q^3)= {q^2(q^2; q^2)_\infty^6 (q^{12}; q^{12})^4_\infty \over(q ;q)^2_\infty (q^3; q^3)^2_\infty (q^4;q^4)^4_\infty (q^6;q^6)_\infty}, \] where \[ \psi(q)= \sum^\infty_{n=0} {(-1)^n q^{(n+1)^2} (q;q^2)_n \over(-q;q)_{2n+1}} \] and \[ \varphi(q)= \sum^\infty_{n=0} {q^{n+1}(-q;q)_{2n} \over(q;q^2)^2_{n+1}}. \] Here \(\psi\) is a sixth-order mock theta function, but \(\varphi\) is not. The other three results involve definite integrals, (Mordell integrals), which are evaluated in terms of the mock theta functions \(\rho,\psi,\lambda\), and infinite products.