How can I calculate the following integration $\int \frac {1}{(y^2 - 10y + 25) ^{ 20}}$?

Question

How can I calculate the following integration $$\int \frac {1}{(y^2 - 10y + 25) ^ {20}}$$? If it were without the power of 20 it can be easily calculated with partial fractions method, but now I can not solve it, any hint will be appreciated.

Answer 1

Start by recognising the perfect square in the denominator.

The integral is in fact $\displaystyle \int \frac 1{(y-5)^{40}}dy$

A simple $u = y-5$ substitution does the trick.

Answer 2

Hint: Use $y^2-10y+25=(y-5)^2$

Answer 3

Hint: Complete the square, then do a $u$-substitution.