How do you integrate #int 1/sqrt(4x+8sqrtx+12) # using trigonometric substitution?

Answer 1

#sqrt(x+2sqrtx+3)-ln(abs(sqrtx+sqrt(x+2sqrtx+3)+1))+C#

We have:

#intdx/sqrt(4x+8sqrtx+12)#
Factor #sqrt4=2# from the denominator:
#=1/2intdx/sqrt(x+2sqrtx+3)#
Complete the square in the denominator. Think about it in terms of #x^2+2x+3#, and then switch #x# terms to #sqrtx# and #x^2# to #x#:
#=1/2intdx/sqrt((sqrtx+1)^2+2)#
Now, let #sqrtx+1=sqrt2tantheta#. Note that this implies that #dx/(2sqrtx)=sqrt2sec^2thetad theta#.

Rearranging some:

#=int(sqrtxdx)/(2sqrtxsqrt((sqrtx+1)^2+2))#
Now, performing the substitutions. We have #dx/(2sqrtx)=sqrt2sec^2thetad theta# present, as well as #sqrtx=sqrt2tantheta-1# in the numerator. Don't forget the switch in the radical in the denominator either:
#=int((sqrt2tantheta-1)sqrt2sec^2thetad theta)/sqrt(2tan^2theta+2)#
Factoring #sqrt2# from the denominator and using #sqrt(tan^2theta+1)=sectheta#:
#=int((sqrt2tantheta-1)sqrt2sec^2theta)/(sqrt2sectheta)d theta#
#=int(sqrt2tantheta-1)secthetad theta#
#=sqrt2inttanthetasecthetad theta-intsecthetad theta#

Both of which are common integrals:

#=sqrt2sectheta-ln(abs(sectheta+tantheta))#
Write #sectheta# in terms of #tantheta#:
#=sqrt2sqrt(tan^2theta+1)-ln(abs(sqrt(tan^2theta+1)+tantheta))#
Using #tantheta=(sqrtx+1)/sqrt2#:
#=sqrt2sqrt((sqrtx+1)^2/2+1)-ln(abs(sqrt((sqrtx+1)^2/2+1)+(sqrtx+1)/sqrt2))#
#=sqrt2sqrt(((sqrtx+1)^2+2)/2)-ln(abs(sqrt(((sqrtx+1)^2+2)/2)+(sqrtx+1)/sqrt2))#
#=sqrt((sqrtx+1)^2+2)-ln(abs((sqrt((sqrtx+1)^2+2)+sqrtx+1)/sqrt2))#
Note that the #1/sqrt2# can actually be taken from the denominator of the logarithm as #-lnsqrt2#, which will be absorbed into the constant of integration in the next step:
#=sqrt(x+2sqrtx+3)-ln(abs(sqrtx+sqrt(x+2sqrtx+3)+1))+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 ( \int \frac{1}{\sqrt{4x + 8\sqrt{x} + 12}} ) using trigonometric substitution:

  1. Recognize that the expression inside the square root resembles a perfect square trinomial.
  2. Let ( u = \sqrt{x} + 2 ).
  3. Square both sides to get ( u^2 = x + 4\sqrt{x} + 4 ).
  4. Rewrite the integral in terms of ( u ) and ( du ).
  5. Simplify the expression inside the square root.
  6. Perform a substitution using trigonometric functions.
  7. Evaluate the integral in terms of the trigonometric substitution.
  8. Back-substitute using the original variable ( x ).
  9. Simplify the result.

The final result will be the integrated expression.

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 3

To integrate (\int \frac{1}{\sqrt{4x+8\sqrt{x}+12}}) using trigonometric substitution, you can make the substitution (x = (\frac{t}{2})^2), which implies (dx = \frac{t}{2} dt). After substitution, you'll have an integral in terms of (t). Then, you can use trigonometric substitution to further simplify the integral.

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