How do you integrate #int 1/sqrt(x^2-16x+3) # using trigonometric substitution?

Answer 1

#I=log|(x-8)+sqrt(x^2-16x+3)|+C#, where,
#C=-logsqrt(61)+c#.
NOTE : -It is better to use #int1/sqrt(X^2+k)dX=ln|x+sqrt(x^2+k)|+c#, from (A)

#I=int1/sqrt(x^2-16x+3)dx=int1/sqrt(x^2-16x+64-61)dx =int1/sqrt((x-8)^2-(sqrt61)^2)dx,.(A)# We take,#x-8=sqrt(61)sectheta=>x=8+sqrt(61)sectheta=>dx=sqrt(61)secthetatantheta*d(theta)# #I=int((sqrt(61)secthetatantheta))/sqrt(61sec^2theta-61)d(theta)# #I=int((sqrt(61)secthetatantheta))/(sqrt(61)sqrt(sec^2theta-1))d(theta)##=int(secthetatantheta)/(tantheta)d(theta)=intsecthetad(theta)##=log|sectheta+tantheta|+c# #I=log|sectheta+sqrt(sec^2theta-1)|+c# #I=log|(x-8)/sqrt(61)+sqrt(((x-8)/sqrt61)^2-1)|+c# #I=log|(x-8)/sqrt(61)+sqrt((x-8)^2-61)/sqrt(61)|+c# #I=log|(x-8)+sqrt(x^2-16x+3)|-log|sqrt(61)|+c#
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Answer 2

Use the substitution #x-8=sqrt61sectheta#.

Let

#I=int1/sqrt(x^2-16x+3)dx#

Complete the square in the square root:

#I=int1/sqrt((x-8)^2-61)dx#
Apply the substitution #x-8=sqrt61sectheta#:
#I=int1/(sqrt61tantheta)(sqrt61secthetatanthetad theta)#

Simplify:

#I=intsecthetad theta#

Integrate directly:

#I=ln|sectheta+tantheta|+C#
Rescale #C#:
#I=ln|sqrt61sectheta+sqrt61tantheta|+C#

Reverse the substitution:

#I=ln|x-8+sqrt(x^2-16x+3)|+C#
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Answer 3

To integrate ( \int \frac{1}{\sqrt{x^2 - 16x + 3}} ) using trigonometric substitution, let ( x = 8 + \sqrt{3} \sec(\theta) ). Then differentiate ( x ) with respect to ( \theta ) and substitute into the integral. This leads to ( dx = \sqrt{3} \sec(\theta) \tan(\theta) d\theta ). Substitute ( x ) and ( dx ) into the integral, simplify, and then use trigonometric identities to express the integrand in terms of trigonometric functions. Finally, evaluate the integral using trigonometric identities and simplify the result.

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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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