How do you integrate #int 1/sqrt(4x^2-12x+4) # using trigonometric substitution?

Answer 1

#int 1/sqrt(4x^2-12x+4) *d x=1/2l n(sqrt(((2x-3)^2)/5-1)+(2x-3)/sqrt 5)+C#

#int 1/sqrt(4x^2-12x+4) *d x=?#
#int 1/sqrt(4x^2-12x+4) *d x=int 1/sqrt(4(x^2-3x+1) )*d x=#
#int 1/sqrt(4x^2-12x+4) *d x=1/2 int 1/sqrt(x^2-3x+1) *d x=#
#x^2-3x+1=(x-3/2)^2-5/4" (complete the square)"#
#int 1/sqrt(4x^2-12x+4) *d x=1/2 int 1/(sqrt((x-3/2)^2-5/4) ) *d x#
#int 1/sqrt(4x^2-12x+4) *d x=1/2 int 1/(sqrt((2x-3)^2/4-5/4)) *d x#
#int 1/sqrt(4x^2-12x+4) *d x=cancel(2)*1/cancel(2) int 1/(sqrt((2x-3)^2-5)) *d x#
#int 1/sqrt(4x^2-12x+4) *d x=int 1/sqrt(5((2x-3)/sqrt 5)^2-1) *d x#
#int 1/sqrt(4x^2-12x+4) *d x=int (sqrt 5/5 ) /sqrt(((2x-3)/(sqrt 5))^2-1) *d x#
#u=(2x-3)/sqrt 5" ; "d u=(2*sqrt5)/5 * d x#
#int 1/sqrt(4x^2-12x+4) *d x=1/2 int(2*sqrt 5/5*d x)/(sqrt(((2x-3)/(sqrt 5))^2-1) )#
#int 1/sqrt(4x^2-12x+4) *d x=1/2 int (d u)/sqrt(u^2-1)#
#int (d u)/sqrt (u^2-1)=l n sqrt(u^2-1)+u#
#"undo substitution"#
#int 1/sqrt(4x^2-12x+4) *d x=1/2l n(sqrt(((2x-3)/sqrt 5)^2-1)+(2x-3)/sqrt 5)#
#int 1/sqrt(4x^2-12x+4) *d x=1/2l n(sqrt(((2x-3)^2)/5-1)+(2x-3)/sqrt 5)+C#
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Answer 2

To integrate ( \int \frac{1}{\sqrt{4x^2 - 12x + 4}} ) using trigonometric substitution, follow these steps:

  1. Complete the square inside the square root.
  2. Make a trigonometric substitution to simplify the integral.
  3. Integrate with respect to the new variable.
  4. Convert back to the original variable.

Let's go through the steps:

  1. Complete the square inside the square root: [ 4x^2 - 12x + 4 = 4(x^2 - 3x + 1) = 4((x - \frac{3}{2})^2 - \frac{1}{4}) ]

  2. Make a trigonometric substitution: [ x - \frac{3}{2} = \frac{1}{2} \sec(\theta) ]

  3. Substitute ( x ) and ( dx ) in terms of ( \theta ) and ( d\theta ): [ x = \frac{1}{2} \sec(\theta) + \frac{3}{2} ] [ dx = \frac{1}{2} \sec(\theta) \tan(\theta) d\theta ]

  4. Substitute the expressions for ( x ) and ( dx ) into the integral and simplify. [ \int \frac{1}{\sqrt{4x^2 - 12x + 4}} , dx = \int \frac{\frac{1}{2} \sec(\theta) \tan(\theta)}{\sqrt{4\left(\frac{1}{2} \sec(\theta) + \frac{3}{2}\right)^2 - 4}} , d\theta ]

  5. Simplify the expression under the square root and integrate with respect to ( \theta ).

  6. After integrating, convert back to the original variable ( x ).

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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.

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