Calculus and Differential Equations Quiz

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.

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.

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).

Laplace transforms are used to transform a function of time into a function of a complex variable s. An example of its application is solving the linear differential equation $y'' - 2y' + 2y = 2e^{-t}$, where the Laplace transform of the equation leads to an algebraic equation in the s-domain.

A function can be represented by a Fourier series in the interval (-π, π) if it is periodic with period 2π and piecewise continuous in that interval.