Finding the antiderivative

In summary: Since the square root is of a difference, you should be thinking of the identity \sin^2\theta + \cos^2\theta = 1. In this case, the difference is of the form 1 - something^2, which means the substitution x - 1 = \sin\theta makes sense.In summary, the student is trying to find the area under the curve of a function and needs to find the antiderivative. They attempt to use the chain rule but realize it is not the correct approach. Instead, they should use trig substitution after completing the square.
  • #1
nhmllr
185
1

Homework Statement


As part of a bigger problem, I am trying to find the area under the curve
f(x) = sqrt{2x - x2} between x = 0 and x = 2. To do this, I have to find the antiderivative of f(x)

Homework Equations


antiderivative
f(x) = axb
F(x) = a/(b+1) * xb+1

chain rule
f(x) = a(b(x))
f'(x) = b'(x) * a'(b(x))

The Attempt at a Solution


I took the derivative of the terms in the parenthesis, then what was outside the parenthesis, to get
2/3 * (x2 - x3/3)3/2
I don't think this is right, because you can use the chain rule to find the derivative, and it's not the original function.

NOTE- I'm not actually old enough and am not actual in a calculus course right now, but I figure that the homework forum would be the best place to put this. Also, don't give me answer, but rather gove me a direction to go in.
 
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  • #2
The best approach, short of looking at a table of integrals, is trig substitution, which you might not have seen yet. Before doing the substitution you need to do some work first, by completing the square.

[tex]\sqrt{2x - x^2} = \sqrt{-(x^2 - 2x)} = \sqrt{-(x^2 - 2x + 1) + 1} = \sqrt{1 - (x - 1)^2}[/tex]

The next step is picking the right trig substitution.
 

What is an antiderivative?

An antiderivative is the inverse operation of a derivative. It is a function that, when differentiated, gives the original function. In simpler terms, it is the function that "undoes" the process of differentiation.

Why is finding the antiderivative important?

Finding the antiderivative allows us to determine the original function from its derivative. This is useful in various areas of science, such as physics and engineering, where the original function represents a physical quantity.

What are the methods for finding an antiderivative?

There are several methods for finding an antiderivative, including the power rule, substitution, integration by parts, and partial fractions. Each method is useful for different types of functions and can be used in combination to find the antiderivative of more complex functions.

What is the constant of integration?

The constant of integration is a constant term that is added to the antiderivative of a function. This constant allows for all possible antiderivatives to be represented, as they differ by a constant. In other words, the constant of integration accounts for the "plus C" that is often seen in antiderivative solutions.

Can all functions have an antiderivative?

No, not all functions have an antiderivative. The function must be continuous and differentiable on its entire domain in order for an antiderivative to exist. Some functions, such as the Dirichlet function, do not have an antiderivative.

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