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.