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Dear Ms. Chester,
Thank you for posting your request on History Hub!
We searched the National Archives Catalog and located a series titled Carded Records Showing Military Service of Soldiers Who Fought in Volunteer Organizations During the American Civil War in the Records of the Adjutant General's Office, 1762-1917 (Record Group 94) that included cards for Mitchem, Isaac M - Age [Blank], Year: [BLANK] - First Cavalry - West Virginia and Mitchem, Isaac N - Age 26, Year: 1862 - First Cavalry - West Virginia. We also located a series titled Regimental and Company Books of Civil War Volunteer Union Organizations, 1861 - 1867 in Record Group 94 that includes the Regimental and Company Books of the 1st West Virginia Cavalry Regiment.
In addition, we searched the National Archives Catalog and located a series titled General Index to Civil War and Later Pension Files in the Records of the Department of Veterans Affairs (Record Group 15). Since the digital images have not been added to the Catalog yet, we searched in Ancestry and located an index card for Rebecca Mitchum (Application No. 306142). There may be a fee for using Ancestry. Instead, you may view these records online at one NARA's facilities for free via a NARA PC. For the nearest NARA location, please consult our web page at https://www.archives.gov/locations/. Also, some of the images from Ancestry are available for free on Family Search at https://www.familysearch.org/en/.
The actual pension file may be located in one of these seres: Case Files of Disapproved Pension Applications of Widows and Other Dependents of Veterans of the Army and Navy, 1861-1934; Case Files of Disapproved Pension Applications of Veterans of the Army and the Navy Who Served Mainly in the Civil War and the War With Spain, 1861 - 1934; or Case Files of Pension Applications Based on Service Completed in the Years 1817 to Approximately 1903 also in Record Group 15. For access to and/or copies of a file that pertains to Rebecca Mitchum, please contact the National Archives at Washington, DC - Textual Reference (RDT1) via email at firstname.lastname@example.org.
We hope this is helpful. Best of luck with you research!