Glossary
Antiderivatives
A function whose derivative is the original function; the process of finding an antiderivative is called integration.
Example:
The antiderivative of 2x is x² + C, because the derivative of x² + C is 2x.
Back-Substitute
The final step in evaluating an indefinite integral using u-substitution, where the new variable *u* is replaced with its original expression in terms of the original variable.
Example:
After integrating ∫cos(u) du to get sin(u) + C, we back-substitute u = x² to get the final answer sin(x²) + C.
Chain Rule
A rule used to find the derivative of a composite function, stating that the derivative of f(g(x)) is f'(g(x)) * g'(x). U-substitution is considered its reverse.
Example:
Using the Chain Rule, the derivative of sin(x²) is cos(x²) * 2x.
Definite Integrals
Integrals that have upper and lower limits of integration, resulting in a numerical value representing the net signed area under a curve over a specific interval.
Example:
Calculating the exact area under the curve of f(x) = x² from x=0 to x=2 requires evaluating a definite integral.
Indefinite Integrals
Integrals that do not have limits of integration and result in a family of functions, represented by an antiderivative plus a constant of integration, C.
Example:
Finding the general antiderivative of f(x) = 3x² involves computing an indefinite integral, which yields x³ + C.
Inner Function
In the context of u-substitution, it is the part of a composite function within the integrand that is chosen to be the new variable, *u*, to simplify the integral.
Example:
In the integral ∫cos(x²) * 2x dx, x² is the inner function that we would typically set as u.
Integration using substitution (u-substitution)
A powerful technique to simplify complex integrals by introducing a new variable, *u*, to transform the integral into a more recognizable and easier-to-evaluate form.
Example:
To evaluate ∫2xcos(x²)dx, we use u-substitution by letting u = x².
Limits of Integration
The upper and lower bounds of a definite integral, which define the interval over which the function is being integrated. When using u-substitution, these limits must be transformed to match the new variable *u*.
Example:
For the integral ∫₁² (2x)/(x²+1)² dx, 1 and 2 are the original limits of integration.