How Can I Integrate x*sqrt(1-x^4) Using Trig Substitution?

Click For Summary
SUMMARY

The integral $$\int x \cdot \sqrt{1-x^4}dx$$ can be effectively solved using the substitution method. By letting $$u = x^2$$, the differential $$du = 2x dx$$ simplifies the integral to $$\dfrac{1}{2} \int \sqrt{1-u^2} du$$. This integral can then be approached using trigonometric substitution, specifically by setting $$u = \sin(w)$$. This method streamlines the integration process and leads to a solution.

PREREQUISITES
  • Understanding of integral calculus
  • Familiarity with substitution methods in integration
  • Knowledge of trigonometric identities
  • Experience with differential calculus
NEXT STEPS
  • Study trigonometric substitution techniques in integration
  • Learn how to apply the Pythagorean identity in integrals
  • Explore advanced integration techniques such as integration by parts
  • Practice solving integrals involving square roots and polynomials
USEFUL FOR

Students studying calculus, mathematics educators, and anyone looking to enhance their skills in solving integrals using substitution methods.

annie122
Messages
51
Reaction score
0
how do i go about integrating

x*sqrt(1-x^4)??

i have no idea
 
Physics news on Phys.org
Re: another trig substitution

Yuuki said:
how do i go about integrating

$$\int x \cdot \sqrt{1-x^4}dx$$

i have no idea

I would let $$u = x^2 \therefore du = 2x dx$$

$$\int x \cdot \sqrt{1-x^4} \cdot \dfrac{du}{2x} = \dfrac{1}{2} \int \sqrt{1-u^2} du$$

Do you know a trig sub for this integral?
 
Re: another trig substitution

u = sin(w) right?

thx!
 
Re: another trig substitution

Yuuki said:
u = sin(w) right?

thx!

Yes, that's the right sub to use. Are you ok going forward?
 
yup thank you
 

Similar threads

Undergrad Different Substitution for Solving an Integral
  • · Replies 6 ·
Replies
6
Views
2K
High School Integration by parts of inverse sine, a solved exercise, some doubts...
  • · Replies 4 ·
Replies
4
Views
3K
MHB How to Evaluate the Integral of an Even Trig Function in a Radical?
  • · Replies 9 ·
Replies
9
Views
1K
Undergrad Substitution in a definite integral
  • · Replies 6 ·
Replies
6
Views
3K
Undergrad Integration Using Trigonometric Substitution
  • · Replies 8 ·
Replies
8
Views
2K
MHB 7.3.5 Integral with trig substitution
  • · Replies 1 ·
Replies
1
Views
2K
MHB Integral using trig substitution
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad What's my mistake in this integration problem?
  • · Replies 8 ·
Replies
8
Views
3K
MHB Integral using trig substitution
  • · Replies 5 ·
Replies
5
Views
2K
High School Trigonometric substitution, a case I'd like to share
  • · Replies 3 ·
Replies
3
Views
3K