Homework Help: Find a and b

1. Jun 25, 2013

utkarshakash

1. The problem statement, all variables and given/known data
If $\displaystyle \int^b_a \dfrac{x^n dx}{x^n + (16-x)^n} = 6$ and a+b=16, then find a and b.

3. The attempt at a solution

$\displaystyle \int^b_a \dfrac{dx}{1 + (16/x - 1)^n} = 6$

I tried substitution but it did not work.

2. Jun 25, 2013

Curious3141

What sub did you try? Hint: try $x = 16-y$ on the original integral.

This is a simple algebra problem, more or less. Very little actual integration need be done.

3. Jun 25, 2013

pasmith

Hint:
$$\int_a^b\frac{x^n}{x^n + (16-x)^n}\,\mathrm{d}x = \int_a^b 1\,\mathrm{d}x - \int_a^b\frac{(16-x)^n}{x^n + (16-x)^n}\,\mathrm{d}x$$
Can you find a substitution which turns the integrand of the second integral on the right into the integrand of the integral on the left?

4. Jun 26, 2013

sankalpmittal

Why do you need substitution right now ?

Apply the property:

I= $\displaystyle \int^b_a \dfrac{x^n dx}{x^n + (16-x)^n}$

And I= $\displaystyle \int^b_a \dfrac{(16-x)^n dx}{(16-x)^n + (16-(16-x))^n}$

Now add the two to get,

2I= ab∫dx

Can you proceed? You already know that I=6...

5. Jun 26, 2013

utkarshakash

Oh that was so easy. I first thought of applying property but somehow couldn't notice that a+b=16 was already given.