How do you integrate #int sqrt(1-x^2)# by trigonometric substitution?

Answer 1

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

Let # I = int sqrt(1-x^2)dx #

The way to approach these types of problems is try and make a comparison with a known trig identity. In the case we will use:

# sin^2A + cos^2A -=1 => cos^2A=1-sin^2A #

If you compare to the integrand, we make the following substitution:

Let #x=sintheta => 1-sin^2theta=cos^2theta# # :. 1-x^2 = cos^2theta # # :. sqrt(1-x^2) = costheta #
And #dx/(d theta) = costheta#
Substituting into the integral we have: # I = int cos theta cos theta d theta # = # :. I = int cos^2 theta d theta #
Now #cos2A-=cos^2A-sin^2A = 2cos^2A-1 # # :. cos^2A=1/2( 1+cos2A)#
And so; # I = int 1/2( 1+cos2theta) d theta # # 2I = int (1+cos2theta) d theta # # 2I = theta + (sin2theta)/2 + C'#
We now need to use the identity #sin2A-=2sinAcosA# # 2I = theta + sinthetacostheta + C'# # :. 2I = sin^-1x + xsqrt(1-x^2) + C'# # :. I = (sin^-1x + xsqrt(1-x^2) + C')/2# # :. I = (sin^-1x + xsqrt(1-x^2))/2 + 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 ∫√(1 - x^2) dx using trigonometric substitution, let x = sin(θ). Then, dx = cos(θ) dθ. Substituting these into the integral gives:

∫√(1 - sin^2(θ)) * cos(θ) dθ

Simplify the expression under the square root:

√(1 - sin^2(θ)) = √(cos^2(θ)) = cos(θ)

So, the integral becomes:

∫cos^2(θ) dθ

Now, use the trigonometric identity cos^2(θ) = (1 + cos(2θ))/2:

∫(1 + cos(2θ))/2 dθ

Integrate term by term:

∫(1/2 + cos(2θ)/2) dθ = (1/2)∫dθ + (1/2)∫cos(2θ) dθ

Integrate each term:

(1/2)θ + (1/4)sin(2θ) + C

Now, substitute back for θ:

(1/2)sin^(-1)(x) + (1/4)sin(2* sin^(-1)(x)) + C

Using the double angle identity sin(2θ) = 2sin(θ)cos(θ), we get:

(1/2)sin^(-1)(x) + (1/4)sin(sin^(-1)(x))cos(sin^(-1)(x)) + C

Remembering that sin^(-1)(x) is the same as arcsin(x), the final answer is:

(1/2)sin^(-1)(x) + (1/4)x√(1 - x^2) + 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 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