How do you integrate #(1-x^2)^.5#?

Answer 1

#int (1-x^2)^0.5 dx = 1/2 x sqrt(1-x^2) + 1/2 sin^(-1)x + C#

Let #x = sin theta#

Then:

#int (1-x^2)^0.5 dx = int (1-sin^2 theta)^0.5 ((d sin theta)/(d theta)) d theta#
#color(white)(int (1-x^2)^0.5 dx) = int (cos^2 theta)^0.5 (cos theta) d theta#
#color(white)(int (1-x^2)^0.5 dx) = int cos^2 theta d theta#
#color(white)(int (1-x^2)^0.5 dx) = int 1/2(cos 2 theta + 1) d theta#
#color(white)(int (1-x^2)^0.5 dx) = 1/4 sin 2 theta + theta/2 + C#
#color(white)(int (1-x^2)^0.5 dx) = 1/2 sin theta cos theta + theta/2 + C#
#color(white)(int (1-x^2)^0.5 dx) = 1/2 x sqrt(1-x^2) + 1/2 sin^(-1)x + C#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To integrate ( \sqrt{1 - x^2} ), you can use the trigonometric substitution method.

Let ( x = \sin(\theta) ), then ( dx = \cos(\theta) , d\theta ).

Substitute ( x = \sin(\theta) ) into ( \sqrt{1 - x^2} ):

[ \sqrt{1 - x^2} = \sqrt{1 - \sin^2(\theta)} = \sqrt{\cos^2(\theta)} = \cos(\theta) ]

Now, integrate ( \cos(\theta) , d\theta ):

[ \int \cos(\theta) , d\theta = \sin(\theta) + C ]

Substitute back ( x = \sin(\theta) ) to get the final result:

[ \int \sqrt{1 - x^2} , dx = \sin(\theta) + C ]

Remembering that ( x = \sin(\theta) ), you can write:

[ \int \sqrt{1 - x^2} , dx = \sin^{-1}(x) + C ]

So, the integral of ( \sqrt{1 - x^2} ) is ( \sin^{-1}(x) + C ), where ( C ) is the constant of integration.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7