[Ethereum] Failing Transaction within Truffle Testing

You can use OpenZeppelin's expectThrow helper -

Source: https://github.com/OpenZeppelin/zeppelin-solidity/blob/master/test/helpers/expectThrow.js

export default async promise => {
      try {
        await promise;
      } catch (error) {
        // TODO: Check jump destination to destinguish between a throw
        //       and an actual invalid jump.
        const invalidJump = error.message.search('invalid JUMP') >= 0;
        // TODO: When we contract A calls contract B, and B throws, instead
        //       of an 'invalid jump', we get an 'out of gas' error. How do
        //       we distinguish this from an actual out of gas event? (The
        //       testrpc log actually show an 'invalid jump' event.)
        const outOfGas = error.message.search('out of gas') >= 0;
        assert(
          invalidJump || outOfGas,
          "Expected throw, got '" + error + "' instead",
        );
        return;
      }
      assert.fail('Expected throw not received');
    };

I use it my test cases like this -

import expectThrow from './helpers/expectThrow';
.
.
.
describe('borrowBook', function() {
        it("should not allow borrowing book if value send is less than 100", async function() {
            await lms.addBook('a', 'b', 'c', 'e', 'f', 'g');
            await lms.addMember('Michael Scofield', accounts[2], "Ms@gmail.com");
            await lms.borrowBook(1, {from: accounts[2], value: 10**12})
            await expectThrow(lms.borrowBook(1, {from: accounts[2], value: 10000})); // should throw exception
        });
});

If they are internal functions you can inherit them and test them, if they are private functions, I believe the only way to unit test them is to make them public/internal, test them and then change them back to private once the tests are passing.

Another way to go about it would be to refactor your code so that the private functions are part of a library that you import into the contract. That way the functions are public in the library but not exposed in the actual contract.