Question

Evaluate the integral, assuming y>2 :
∫√(y²-4)/y⁵ d y=

Answer 1

The integral \( \int \sqrt{y^2 - 4}/y^5 dy \) assuming y>2 is solved using a substitution method, with u = y^2 - 4, to simplify the integral into a solvable form, and then finding the antiderivative to get the solution in terms of y after simplification.

Step-by-step explanation:

The problem is asking to evaluate the integral \( \int \sqrt{y^2 - 4}/y^5 dy \), assuming that y>2. To solve this integral, we can use a substitution method where we let u = y^2 - 4, so that du = 2y dy. This simplifies the integral to \( \int u^{-1/2} * 2u^{+1} du \) which is an easily solvable form. After integrating, we convert back to terms of y to find the solution.

To find the antiderivative, we can use the following steps:

1. Let u = y^2-4, hence du = 2y dy.
2. Express dy in terms of du which gives dy = du/(2y).
3. Rewrite the integral in terms of u which is \( \int u^{-1/2} (u+4)^{-5/2} * (1/2) du \).
4. Integrate using standard integral formulas for powers of u.
5. Replace u with y^2-4 to express the answer back in terms of the original variable y.

The final answer is obtained after simplification.