For the following integral find an appropriate TRIGONOMETRIC SUBSTITUTION of the form x=f(t) to simplify the integral.

INT (x)/(sqrt(-191-8x^2+80x))dx

x=?

The integral will be of the form

(y+5)/sqrt(a^2-y^2)
if you complete the square of the expression under the square-root sign using
y=x-5, or x=y-5
That will reduce the given integral to two simpler ones.

sint

To find an appropriate trigonometric substitution, we start by examining the expression inside the square root: -191 - 8x^2 + 80x.

We notice that this expression contains a quadratic term (-8x^2) and a linear term (80x). To simplify it, we can complete the square.

First, let's factor out a common factor of -8:

-8(x^2 - 10x + 23.75)

Now, to complete the square, we add and subtract the square of half the linear coefficient (10/2 = 5):

-8(x^2 - 10x + 25 - 25 + 23.75)

Simplifying further:

-8((x - 5)^2 - 1.25)

Now, we can rewrite the expression as:

-8(x - 5)^2 + 10

Replacing this expression in the original integral:

∫ (x) / (√(-191 - 8x^2 + 80x)) dx

∫ (x) / (√(-8(x - 5)^2 + 10)) dx

Now, we can see that a trigonometric substitution of the form x = f(t) = 5 + √(10)sin(θ) will simplify the integral.

Let's find dx in terms of dθ:

If x = 5 + √(10)sin(θ),

then dx = √(10)cos(θ) dθ

Now, substitute x and dx in the integral:

∫ ((5 + √(10)sin(θ))) / (√(-8((5 + √(10)sin(θ)) - 5)^2 + 10)) (√(10)cos(θ)) dθ

Simplifying this expression will give you the appropriate trigonometric substitution.