integrand

Examples of integrand in the following topics:

• Trigonometric Substitution

  • Trigonometric functions can be substituted for other expressions to change the form of integrands.
  • If the integrand contains $a^2+x^2$, let $x = a \tan(\theta)$ and use the identity:
  • If the integrand contains $x^2-a^2$, let $x = a \sec(\theta)$ and use the identity:
  • Example 1: Integrals where the integrand contains $a^2 − x^2$ (where $a$ is positive)
  • Example 2: Integrals where the integrand contains $a^2 − x^2$ (where $a$ is not zero)

• Numerical Integration

  • The integrand $f(x)$ may be known only at certain points, such as obtained by sampling.
  • A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function.
  • Numerical integration methods can generally be described as combining evaluations of the integrand to get an approximation to the integral.
  • Also, each evaluation takes time, and the integrand may be arbitrarily complicated.
  • A 'brute force' kind of numerical integration can be done, if the integrand is reasonably well-behaved (i.e. piecewise continuous and of bounded variation), by evaluating the integrand with very small increments.

• Trigonometric Integrals

• Numerical Integration

  • The integrand $f(x)$ may be known only at certain points, such as when obtained by sampling.
  • A formula for the integrand may be known, but it may be difficult or impossible to find an antiderivative which is an elementary function.
  • An example of such an integrand $f(x)=\exp(-x^2)$, the antiderivative of which (the error function, times a constant) cannot be written in elementary form.

• Improper Integrals

  • Integrals are also improper if the integrand is undefined at an interior point of the domain of integration, or at multiple such points.
  • The problem here is that the integrand is unbounded in the domain of integration (the definition requires that both the domain of integration and the integrand be bounded).

• Cylindrical Shells

  • In the integrand, the factor $x$ represents the radius of the cylindrical shell under consideration, while  is equal to the height of the shell.
  • Therefore, the entire integrand, $2\pi x \left | f(x) - g(x) \right | \,dx$, is nothing but the volume of the cylindrical shell.

• Indefinite Integrals and the Net Change Theorem

  • $f(x)$, the function being integrated, is known as the integrand.

• Inverse Compton Spectra - Single Scattering

  • Depending on the value of $E_f/E_0$ this integral may vanish.Specifically the integrand is non-zero only if $\mu_f$ lies in the range

• Absolute Convergence and Ratio and Root Tests

  • Similarly, an improper integral of a function, $\textstyle\int_0^\infty f(x)\,dx$, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if $\int_0^\infty \left|f(x)\right|dx = L$.

• The Physics of Bremsstrahlung

  • If $\omega \tau \gg\ 1$, the integrand will oscillate rapidly so the integral will be small.