Just have an object as a member:
class DBUser
{
private:
SQLite::Database db;
public:
explicit DBUser(std::string &path) :
db (path, SQLITE_OPEN_READWRITE|SQLITE_OPEN_CREATE)
{
}
void test() {
db.exec("DROP TABLE IF EXISTS test");
db.exec("CREATE TABLE test (id INTEGER PRIMARY KEY, foobar TEXT)");
}
}