Learning Trigonometric Substitution

Click For Summary
SUMMARY

The forum discussion focuses on the application of trigonometric substitution in the integral of sin^5(x). The user attempts to solve the integral by breaking it down into sin^4(x)sin(x) and subsequently using the identity (1 - cos^2(x))^2. The final result, -cos(x) + (2/3)cos^3(x) - (1/5)cos^5(x) + C, is achieved through a U substitution where u = cos(x) and du = -sin(x)dx. The discussion highlights the importance of understanding substitution techniques in integral calculus.

PREREQUISITES
  • Understanding of integral calculus and basic integration techniques
  • Familiarity with trigonometric identities and their applications
  • Knowledge of U substitution method in integration
  • Ability to manipulate algebraic expressions involving trigonometric functions
NEXT STEPS
  • Study U substitution in depth, focusing on its application in various integrals
  • Explore trigonometric identities and their proofs for better integration techniques
  • Practice solving integrals involving higher powers of sine and cosine functions
  • Learn about integration by parts and its relationship with trigonometric functions
USEFUL FOR

Students studying calculus, particularly those focusing on integral techniques, as well as educators seeking to clarify trigonometric substitution methods in their teaching.

1MileCrash
Messages
1,338
Reaction score
41

Homework Statement



My book has given me this example, with a step by step, but one of the steps have left me dumbfounded, they seem to have left a huge gap in steps..

Homework Equations





The Attempt at a Solution



\int sin^{5}xdx

\int sin^{4}x sinx dx

\int (1-cos^{2}x)^{2} sinx dx

\int (1 - 2cos^{2} x + cos^{4} x) sinx dx

Cool story bro. Here's the next step, what happened?

= - cos x + \frac{2}{3}cos^{3}x - \frac{1}{5}cos^{5} x + C

Did they bring a negative out of the integral and do some kind of U substitution?
 
Last edited by a moderator:
Physics news on Phys.org
They put u = cos(x), du = -sin(x)dx. Try it.
 

Similar threads

Integrate [cosec(30°+x)-cosec(60°+x)] dx in terms of tan x
  • · Replies 3 ·
Replies
3
Views
2K
Use substitution to solve the definite integral
  • · Replies 9 ·
Replies
9
Views
2K
Calculating radius of gyration of plane figure about x-axis
  • · Replies 5 ·
Replies
5
Views
3K
Potential of a rotationally symmetric charge distribution
  • · Replies 4 ·
Replies
4
Views
1K
How Does Substitution Affect Double Integration and Differentiation?
  • · Replies 3 ·
Replies
3
Views
2K
What Are Some Substitutions That Can Be Used to Solve This Integral?
  • · Replies 5 ·
Replies
5
Views
2K
Find the result of definite integration
  • · Replies 6 ·
Replies
6
Views
2K
Integral using substitution x = -u
  • · Replies 12 ·
Replies
12
Views
2K
How should I find the equilibrium points and the general equation?
  • · Replies 3 ·
Replies
3
Views
2K
Integration via Trigonometric Substitution
  • · Replies 11 ·
Replies
11
Views
2K