Examples of integrand in the following topics:
-
- Trigonometric functions can be substituted for other expressions to change the form of integrands.
- If the integrand contains a2+x2, let x=atan(θ) and use the identity:
- If the integrand contains x2−a2, let x=asec(θ) 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)
-
- 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.
-
-
- 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(−x2), the antiderivative of which (the error function, times a constant) cannot be written in elementary form.
-
- 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).
-
- 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, 2πx∣f(x)−g(x)∣dx, is nothing but the volume of the cylindrical shell.
-
- f(x), the function being integrated, is known as the integrand.
-
- Depending on the value of Ef/E0 this integral may vanish.Specifically the integrand is non-zero only if μf lies in the range
-
- Similarly, an improper integral of a function, ∫0∞f(x)dx, is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if ∫0∞∣f(x)∣dx=L.
-
- If ωτ≫ 1, the integrand will oscillate rapidly so the integral will be small.