Periodic inspections can accurately detect hidden failures. They are called perfect but are usually costly. The periods of perfect inspections (PIs) cannot be easily changed due to regulations and management problems. With improvements in monitoring technology, many devices for remote inspections are available at reasonable cost when compared with those for PIs. Remote inspections are called imperfect in that hidden failures can be detected with a probability of less than 100%. The periods of imperfect inspections (IPIs) can be easily adjusted. We propose an integrated inspection policy that combines PIs and IPIs and is flexible in that the frequencies of IPIs need not be common for all PI periods. The inspection frequencies of IPIs are determined to minimize the total expected cost, which includes inspections and penalty cost due to undetected failures. We investigated the property of the total expected cost function for optimization of IPI frequencies and used this obtained property to implement an optimization algorithm. We provide numerical examples to compare the expected costs of the proposed policy with flexible IPI frequencies and a policy with common IPI frequencies. These examples indicate that the proposed policy can reduce the total expected cost for systems with increasing failure rate and enhance the effectiveness of the inspection plan.