For the following integral find an appropriate TRIGONOMETRIC SUBSTITUTION of the form x=f(t) to simplify the integral.
INT x(sqrt(8x^2-64x+120))dx
x=?
You could try the similar technique as:
http://www.jiskha.com/display.cgi?id=1260825344
In this case, it is a multiplication, so the result is slightly different.
However, after completing the squares, you will get the form
let y=x-4, x=y+4
(1/√8)(y+4)*sqrt(y^2-1)
which should still be mangeable.
To find an appropriate trigonometric substitution for the given integral, let's begin by completing the square inside the square root.
The quadratic expression inside the square root, 8x^2 - 64x + 120, can be rewritten as follows:
8x^2 - 64x + 120 = 8(x^2 - 8x + 15)
To complete the square, we need to add and subtract a constant term inside the parentheses that will ensure the expression becomes a perfect square trinomial. The constant term can be found by taking half the coefficient of the linear term (-8) squared:
x^2 - 8x + 15 = (x^2 - 8x + 16 - 1)
Now, we have a perfect square trinomial:
(x - 4)^2 - 1
We can rewrite the integral as:
∫ x * √((x - 4)^2 - 1) dx
Now, let's substitute x - 4 = sech(t).
Taking the derivative of both sides gives: dx = sech(t) * tanh(t) dt.
Substituting these values into the integral, we get:
∫ (sech(t) + 4) * √(sech^2(t) - 1) * sech(t) * tanh(t) dt
Simplifying this expression gives us:
∫ (sech^2(t) + 4sech(t)) * √(sech^2(t) - 1) * sech(t) * tanh(t) dt
At this point, the integral is ready for evaluation using trigonometric identities and integration techniques for trigonometric functions.