#55
(a.) Partial fractions
(b.) Partial fractions
(c.) Trigonometric substitution (use the "completing the square" operation)
As for #53, use polynomial division.
I don't understand #54 at all.
For 53: You want to perform long division first, because the degree of the numerator is greater than that of the denominator.
for 54: a) You need a partial fraction with denominator for each k value between 1 and m. The same applies to part b)
For 55:
a) Multiply the numerator by 2 and divide by 2 before the integral sign. Note that the new numerator is the derivative of the denominator, so the resulting antiderivative is a one half times the logarithm of the denominator... plus a constant, of course.
b) partial fractions ... factor denominator, then split into fractions.
c) substitute
Good luck!