要正确关闭Apache HttpClient连接，可以使用以下代码示例：

import org.apache.http.HttpEntity;
import org.apache.http.client.methods.CloseableHttpResponse;
import org.apache.http.client.methods.HttpGet;
import org.apache.http.impl.client.CloseableHttpClient;
import org.apache.http.impl.client.HttpClients;
import org.apache.http.util.EntityUtils;

public class HttpClientExample {
    public static void main(String[] args) {
        CloseableHttpClient httpClient = HttpClients.createDefault();
        CloseableHttpResponse response = null;
        
        try {
            HttpGet httpGet = new HttpGet("https://example.com/api");
            response = httpClient.execute(httpGet);
            
            // 处理响应
            HttpEntity entity = response.getEntity();
            String responseBody = EntityUtils.toString(entity);
            System.out.println(responseBody);
            
            // 关闭连接
            EntityUtils.consume(entity);
        } catch (Exception e) {
            e.printStackTrace();
        } finally {
            try {
                if (response != null) {
                    response.close();
                }
            } catch (Exception e) {
                e.printStackTrace();
            }
            
            try {
                if (httpClient != null) {
                    httpClient.close();
                }
            } catch (Exception e) {
                e.printStackTrace();
            }
        }
    }
}

在上述示例中，我们使用HttpClients.createDefault()方法创建一个默认的CloseableHttpClient实例，发送HttpGet请求并获取响应。在处理完响应后，我们使用EntityUtils.consume(entity)方法消耗响应实体，然后通过调用response.close()和httpClient.close()来关闭连接。在finally块中进行这些关闭操作可以确保无论发生什么异常，都能正确地关闭连接。