Glossary
Antiderivative of 1/(1+x^2)
The function whose derivative is 1/(1+x^2). It is tan^-1(x) + C (or arctan(x) + C).
Example:
The integral ∫(1/(1+x^2))dx is a classic example of an antiderivative of 1/(1+x^2), yielding arctan(x) + C.
Antiderivative of 1/sqrt(1-x^2)
The function whose derivative is 1/sqrt(1-x^2). It is sin^-1(x) + C (or arcsin(x) + C).
Example:
If you see ∫(1/sqrt(1-x^2))dx, you should recall that the antiderivative of 1/sqrt(1-x^2) is arcsin(x) + C.
Antiderivative of 1/x
The function whose derivative is 1/x. It is ln|x| + C, where the absolute value ensures the domain matches that of 1/x.
Example:
When integrating 3/x, you use the antiderivative of 1/x to get 3ln|x| + C, remembering the absolute value for the domain.
Antiderivative of cos(x)
The function whose derivative is cos(x). It is sin(x) + C.
Example:
The antiderivative of cos(x) is sin(x) + C, which is often used when integrating periodic functions.
Antiderivative of csc(x)cot(x)
The function whose derivative is csc(x)cot(x). It is -csc(x) + C.
Example:
The antiderivative of csc(x)cot(x) is -csc(x) + C, completing the set of basic reciprocal trig function integrals.
Antiderivative of csc^2(x)
The function whose derivative is csc^2(x). It is -cot(x) + C.
Example:
The antiderivative of csc^2(x) is -cot(x) + C, a less common but still important integral to recognize.
Antiderivative of e^x
The function whose derivative is e^x. It is e^x + C.
Example:
The antiderivative of e^x is simply e^x + C, making it one of the easiest integrals to remember.
Antiderivative of sec(x)tan(x)
The function whose derivative is sec(x)tan(x). It is sec(x) + C.
Example:
If you encounter ∫sec(x)tan(x)dx, you immediately know the antiderivative of sec(x)tan(x) is sec(x) + C.
Antiderivative of sec^2(x)
The function whose derivative is sec^2(x). It is tan(x) + C.
Example:
To find the function whose slope is always sec^2(x), you would use the antiderivative of sec^2(x), which is tan(x) + C.
Antiderivative of sin(x)
The function whose derivative is sin(x). It is -cos(x) + C.
Example:
If a particle's acceleration is sin(t), its velocity function would be -cos(t) + C, which is the antiderivative of sin(x).
Antiderivatives
A function F(x) is an antiderivative of f(x) if the derivative of F(x) is f(x). It is the reverse process of differentiation.
Example:
If the velocity of a particle is given by v(t) = 2t, then its position function, s(t) = t^2 + C, is an antiderivative because s'(t) = v(t).
Antiderivatives of Inverse Trig Functions
Integrals that result in inverse trigonometric functions, typically arising from specific fractional forms.
Example:
While less frequent, recognizing the forms that lead to antiderivatives of inverse trig functions, like 1/(1+x^2) for arctan(x), can save time on the exam.
Antiderivatives of Transcendental Functions
Integrals involving functions like e^x and ln|x|, which are not algebraic.
Example:
Mastering the antiderivatives of transcendental functions like e^x and 1/x is essential for a wide range of calculus problems.
Antiderivatives of Trigonometric Functions
The inverse operations of differentiating trigonometric functions, yielding the original function plus a constant.
Example:
Knowing the antiderivatives of trigonometric functions is crucial; for instance, the antiderivative of cos(x) is sin(x) + C.
Constant of Integration (+C)
An arbitrary constant added to the end of every indefinite integral. It accounts for the fact that the derivative of any constant is zero, meaning multiple functions can have the same derivative.
Example:
After integrating 2x, we must add the constant of integration to get x^2 + C, acknowledging that x^2+5 and x^2-10 both have a derivative of 2x.
Definite Integral
An integral with specified upper and lower bounds, which evaluates to a specific numerical value representing the net signed area under a curve.
Example:
Unlike an indefinite integral, a definite integral like ∫ from 0 to 1 of x dx evaluates to a single number, 1/2, representing the area under the curve from x=0 to x=1.
Family of Functions
A set of functions that differ only by a constant. All functions within a family share the same derivative.
Example:
The functions y = x^2 + 1, y = x^2 - 5, and y = x^2 + 100 all belong to the same family of functions because their derivative is 2x.
Indefinite Integral
The general form of all antiderivatives of a function f(x), denoted by ∫f(x)dx = F(x) + C. It represents a family of functions.
Example:
When you calculate ∫(3x^2)dx, the result, x^3 + C, is an indefinite integral representing all functions whose derivative is 3x^2.
Multiples Rule (for Antiderivatives)
This rule states that a constant factor can be moved outside the integral sign: ∫c ⋅ f(x)dx = c ∫f(x)dx.
Example:
To integrate 5e^x dx, the multiples rule lets you pull the 5 out, so you integrate e^x and then multiply by 5, resulting in 5e^x + C.
Reverse Power Rule
A fundamental rule for finding the antiderivative of a power function x^n. It states that ∫x^n dx = (x^(n+1))/(n+1) + C, provided n ≠ -1.
Example:
To find the antiderivative of x^5, apply the reverse power rule to get (x^(5+1))/(5+1) + C, which simplifies to (x^6)/6 + C.
Sums Rule (for Antiderivatives)
This rule states that the integral of a sum of functions is the sum of their individual integrals: ∫[f(x) + g(x)]dx = ∫f(x)dx + ∫g(x)dx.
Example:
When integrating (x^2 + sin(x))dx, the sums rule allows you to integrate x^2 and sin(x) separately and then add their results: (x^3)/3 - cos(x) + C.