How Do You Integrate Trigonometric Functions with Substitution?

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SUMMARY

The discussion focuses on integrating the function \(\int \sin(\pi x)\sqrt{1+2\pi\cos^2(\pi x)}\,dx\). The initial reduction to \(\sqrt{u^2-u}\) is noted, but further steps are required for completion. A recommended substitution is \(u = \cos(\pi x)\), leading to \(du = -\pi\sin(\pi x) \, dx\). This substitution is essential for simplifying the integral and progressing towards the solution.

PREREQUISITES
  • Understanding of trigonometric functions and their properties
  • Familiarity with integration techniques, specifically substitution
  • Knowledge of calculus, particularly integral calculus
  • Ability to manipulate algebraic expressions involving square roots
NEXT STEPS
  • Study the method of substitution in integral calculus
  • Explore integration of trigonometric functions using various techniques
  • Learn about the properties of definite and indefinite integrals
  • Investigate advanced integration techniques, such as integration by parts
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus, as well as anyone looking to enhance their skills in integrating trigonometric functions.

annie122
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how do i integrate sin(pi*x)*sqrt(1 + pi*2*cos(pi*x)^2)?
i reduced this to sqrt(u^2-u) but i don't know how to go from here
 
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Re: integration

I think what you have written should be interpreted as:

$$\int \sin(\pi x)\sqrt{1+2\pi\cos^2(\pi x)}\,dx$$

Is this correct? And if so, what substitution did you use?
 
Re: integration

Yuuki said:
how do i integrate sin(pi*x)*sqrt(1 + pi*2*cos(pi*x)^2)?
i reduced this to sqrt(u^2-u) but i don't know how to go from here

Substitute \displaystyle \begin{align*} u = \cos{ \left( \pi \, x \right) } \implies du = -\pi\sin{ \left( \pi \, x \right) } \, dx \end{align*} to start with...
 
Last edited:

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