usubstitusjon

Usubstitusjon, commonly called u-substitution in English, is a technique in integral calculus used to simplify the evaluation of certain integrals by transforming a composite function into a simpler form. The idea is to replace a chosen inner function g(x) by a new variable u, so that the differential du matches the remaining differential in the integral. This is often described as applying the reverse chain rule.

To perform u-substitution:

- choose u = g(x) such that the integrand contains du,

- compute du = g'(x) dx and rewrite the integral in terms of u,

- integrate with respect to u, then substitute back x in place of u.

Commonly, u-substitution is used when the integrand is a composition f(g(x)) multiplied by g'(x), allowing the integral to become ∫ F(u) du. It is widely employed for polynomial expressions under a square root, exponential, or trigonometric integrals after an inner function is identified.

Example: ∫ 2x cos(x^2) dx. Let u = x^2, du = 2x dx. The integral becomes ∫ cos(u) du = sin(u) + C = sin(x^2) + C.

Limitations: u-substitution requires that the derivative of the inner function is present in the integrand or can be factored out. For definite integrals, limits can be changed to u-values to avoid back-substitution.

In Norwegian mathematics, the method is often referred to as usubstitusjon or u-substitusjon and is one of the basic techniques of integration alongside methods like parts and partial fractions.