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Explain Bernoulli's formula in integral calculus and provide an example of its application.
Explain Bernoulli's formula in integral calculus and provide an example of its application.
Bernoulli's formula in integral calculus is given by $ \int x^n , dx = \frac{x^{n+1}}{n+1} + C$, where $n$ is a constant and $C$ is the constant of integration. An example of its application is $\int x^3 , dx = \frac{x^4}{4} + C.
What are reduction formulae in integral calculus and provide an example?
What are reduction formulae in integral calculus and provide an example?
Reduction formulae in integral calculus are used to reduce the power of a given expression. An example is the reduction formula for $ \int \sin^n x , dx = - \frac{1}{n} \sin^{n-1} x \cos x + \frac{n-1}{n} \int \sin^{n-2} x , dx.
What is Fourier series and under what conditions can it represent a function in the interval (0, 2π)?
What is Fourier series and under what conditions can it represent a function in the interval (0, 2π)?
Fourier series represents a function in the interval (0, 2π) if the function is periodic and piecewise continuous in that interval. It can be represented as $f(x) = \frac{a_0}{2} + \sum_{n=1}^{\infty} (a_n \cos nx + b_n \sin nx).
Explain the concept of Laplace transforms and provide an example of its application in solving a linear differential equation.
Explain the concept of Laplace transforms and provide an example of its application in solving a linear differential equation.
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What are the conditions for a function to be representable by a Fourier series in the interval (-π, π)?
What are the conditions for a function to be representable by a Fourier series in the interval (-π, π)?
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