Integration by substitution, also known as u-substitution, is a method used in calculus to evaluate integrals. It involves substituting a variable in the integrand with a new variable, which allows for the integral to be rewritten in a simpler form.
Integration by substitution is useful when the integrand contains a function within another function, such as f(g(x)). It is also helpful when the integrand contains a polynomial multiplied by a trigonometric or exponential function.
The substitution variable, denoted as u, should be chosen so that it simplifies the integrand and makes it easier to integrate. It is often helpful to choose u as the inner function of the composite function in the integrand.
The general process for integration by substitution involves four steps: 1) Identify the substitution variable u, 2) Rewrite the integrand in terms of u, 3) Replace the differential dx with du in the integral, and 4) Integrate the simplified integral with respect to u and then substitute back in the original variable.
Yes, there are some limitations and restrictions when using integration by substitution. The substitution variable u must be a continuous and differentiable function, and the integrand must also be continuous and differentiable with respect to u. Additionally, the limits of integration may need to be adjusted when making the substitution.