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Dirichlet’s integral and its applications

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Question 1

The value of the Dirichlet’s integral [Tex]\int_0^{\pi} \frac{\sin x}{x} dx[/Tex]

  • 1

  • [Tex]\frac{\pi}{2}[/Tex]

  • 2

  • 0

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Question 2

Dirichlet’s integral is mainly applied to:

  • Compute areas under curves

  • Solve PDEs in quantum mechanics and electrical engineering

  • Solve cubic equations

  • Compute limits of sequences

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Question 3

Dirichlet’s integral is frequently used in:

  • Solving linear equations

  • ourier analysis

  • Solving quadratic equations

  • Probability distributions

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Question 4

The main mathematical idea behind Dirichlet’s integral is:

  • Integration by parts

  • Limiting behavior of oscillatory functions

  • Substitution of variables

  • Use of partial fractions

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Question 5

 In signal processing, the Dirichlet integral helps in:

  • Calculating the Fourier transform of ideal filters

  • Finding eigenvalues of matrices

  • Solving differential equations numerically

  • Estimating probability density functions

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Question 6

For Dirichlet’s integral, which property ensures its convergence?

  • Monotonicity of sin⁡x

  • Oscillatory cancellation of sin⁡x

  • Positivity of sin⁡x

  • Continuity of sin⁡x

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