Calculus with inverse trig functions

In summary, the integral of (1/Sqrt(5x-x^2)) can be evaluated by completing the square and using the equation [d/dx]{arcsin(x)}=(du/dx)/sqrt(1-x^2). The final solution is arcsine(2x-5)/5, but the process of finding this solution involves finding the value of k by completing the square for 5x-x^2.
  • #1
famallama
9
0

Homework Statement


Evaluate the integral of (1/Sqrt(5x-x^2))


Homework Equations


[d/dx]{arcsin(x)}=(du/dx)/sqrt(1-x^2)


The Attempt at a Solution


arcsine(2x-5)/5


I did end up getting the right answer, but have no idea how I got there.
 
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  • #2
So we have

[tex]\int \frac{1}{\sqrt{5x-x^2}}[/tex]

try completing the square for 5x-x2 (i.e. put it into the form a[x+h]2+k)
 
  • #3
I have tried that, but the 5 is what is throwing me off.
 
  • #4
famallama said:
I have tried that, but the 5 is what is throwing me off.

5x-x2 = -(x2-5x) = -(x+5/2)2+k. Can you find 'k'?
 
  • #5
thank you very much
 

1. What is the purpose of using inverse trig functions in calculus?

Inverse trig functions are used in calculus to help solve problems involving angles and triangles. They allow us to find the angle measure given a specific trigonometric ratio, or to find the side length of a triangle when given an angle and another side length.

2. What are the most commonly used inverse trig functions in calculus?

The three most commonly used inverse trig functions in calculus are arcsine (sin-1), arccosine (cos-1), and arctangent (tan-1). These functions are used to find the angle measure given the trigonometric ratio of a right triangle.

3. How do you differentiate inverse trig functions?

To differentiate inverse trig functions, you can use the chain rule. For example, if you have y = sin-1(x), you can rewrite it as y = arcsin(x) and then use the chain rule to find the derivative as dy/dx = 1/√(1-x2).

4. Can you integrate inverse trig functions?

Yes, you can integrate inverse trig functions by using the reverse of the chain rule. For example, if you have y = arcsin(x), you can rewrite it as y = sin-1(x) and then integrate it as ∫dy = ∫dx/√(1-x2). This can be solved using trigonometric substitution or integration by parts.

5. How do you use inverse trig functions to solve real-world problems?

Inverse trig functions can be used to solve real-world problems involving angles and triangles, such as finding the height of a building or the distance between two points. They can also be used in physics and engineering to analyze motion and forces, as well as in navigation and surveying to calculate distances and angles.

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