U-substitution

Definition

U-substitution is a method used in calculus to simplify the process of integration by substituting a new variable for a complicated function. This technique often makes it easier to evaluate integrals and is closely linked to the chain rule, allowing for straightforward computation of derivatives and integrals when dealing with composite functions. By identifying an inner function and substituting it with a single variable, one can transform a complex integral into a more manageable form.

5 Must Know Facts For Your Next Test

1. U-substitution relies on the relationship between a function and its derivative, where substituting 'u' can simplify the integral's computation.
2. When using u-substitution, it's essential to change both the variable and the limits of integration if evaluating a definite integral.
3. To effectively use u-substitution, one should identify an inner function that can be expressed as 'u', making sure to compute its derivative as part of the substitution process.
4. The goal of u-substitution is often to convert the integral into a standard form that is easier to integrate, allowing for simpler calculations.
5. Common applications of u-substitution include integrals involving polynomials, exponential functions, and trigonometric identities where recognizing patterns can lead to successful substitutions.

Review Questions

• How does u-substitution relate to the chain rule in calculus?
  • U-substitution is fundamentally tied to the chain rule because it helps simplify integrals that involve composite functions. When you recognize an inner function within an integral, you can use u-substitution to replace that inner function with a new variable 'u'. This approach mirrors how the chain rule operates when finding derivatives, as both techniques utilize the relationship between functions and their derivatives to facilitate computation.
• What steps must be taken when applying u-substitution in evaluating definite integrals?
  • When applying u-substitution to definite integrals, start by selecting an appropriate substitution for 'u' based on an inner function. After performing the substitution, you'll also need to change the limits of integration to correspond with your new variable 'u'. Once you have set up the integral with these new limits, you can evaluate it as usual before converting back to 'x' if necessary. Finally, remember to ensure that all components, including differential elements, are correctly transformed.
• Evaluate the integral $$\int (3x^2 + 1) e^{x^3 + x} \, dx$$ using u-substitution and explain your reasoning.
  • To evaluate this integral using u-substitution, let's set 'u' equal to the expression inside the exponential: $$u = x^3 + x$$. Then, we compute its derivative: $$du = (3x^2 + 1) dx$$. This means our integral transforms directly into $$\int e^u \, du$$. The integral of $$e^u$$ is simply $$e^u + C$$. Finally, we substitute back our original expression for 'u': $$e^{x^3 + x} + C$$. The reasoning shows how recognizing an appropriate substitution streamlines integration.

Related terms

Chain Rule:

A formula used to compute the derivative of a composite function, stating that the derivative of a function at a point is the product of the derivative of the outer function and the derivative of the inner function.

Definite Integral: An integral that calculates the net area under a curve between two specified limits, providing a numerical value rather than a function.

Differential:

An infinitesimally small change in a variable, which is used in calculus to describe how functions change as their inputs vary.