How do you integrate #int x^2 /sqrt( 16+x^4 )dx# using trigonometric substitution?

Answer 1

This cannot be integrated using elementary functions.

Use the substitution #x^2=4tantheta#. This implies that #2xdx=4sec^2thetad theta#. Also keep in mind that #x=2sqrttantheta#.

We have:

#intx^2/sqrt(16+x^4)dx=1/2int(x(2xdx))/sqrt(16+x^4)#
#=1/2int(2sqrttantheta(4sec^2thetad theta))/sqrt(16+16tan^2theta)=int(sqrttantheta(sec^2theta)d theta)/sqrt(1+tan^2theta)#
Note that #1+tan^2theta=sec^2theta#, so #sectheta=sqrt(1+tan^2theta)#:
#=int(sqrttantheta(sec^2theta)d theta)/sectheta=intsqrttanthetasecthetad theta#

The more we continue, we see that this cannot be integrated using elementary functions.

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{x^2}{\sqrt{16+x^4}} , dx ) using trigonometric substitution:

  1. Recognize that the expression under the square root resembles a sum of squares, so we can use trigonometric substitution.
  2. Let ( x^2 = 4 \tan(\theta) ), which implies ( dx = 2 \tan(\theta) \sec^2(\theta) , d\theta ).
  3. Substitute ( x^2 = 4 \tan(\theta) ) and ( dx = 2 \tan(\theta) \sec^2(\theta) , d\theta ) into the integral.
  4. The integral becomes ( \int \frac{4\tan^2(\theta) \sec^2(\theta)}{\sqrt{16 + 4\tan^2(\theta)}} , d\theta ).
  5. Simplify to get ( \int \frac{4\tan^2(\theta) \sec^2(\theta)}{\sqrt{4(4 + \tan^2(\theta))}} , d\theta ).
  6. Simplify further to ( \int \frac{4\tan^2(\theta) \sec^2(\theta)}{2\sqrt{4 + \tan^2(\theta)}} , d\theta ).
  7. Reduce to ( \int 2 \tan^2(\theta) , d\theta ).
  8. Use trigonometric identity ( \tan^2(\theta) = \sec^2(\theta) - 1 ) to rewrite the integral as ( \int 2(\sec^2(\theta) - 1) , d\theta ).
  9. Integrate term by term to get ( 2 \tan(\theta) - 2\theta + C ).
  10. Substitute back ( \theta = \arctan\left(\frac{x^2}{4}\right) ) to get the final result ( \boxed{2 \tan\left(\arctan\left(\frac{x^2}{4}\right)\right) - 2\arctan\left(\frac{x^2}{4}\right) + 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