An indefinite integral represents a family of antiderivatives, written with +C.
The Fundamental Theorem of Calculus links antiderivatives to definite integrals.
The integral symbol means “find all antiderivatives.”
Definite integrals have limits and produce a number.
Indefinite integrals have no limits and produce a function + C.
Algebraic simplification of the integrand often makes integration easier.
Many integrals come directly from reversing derivative rules.
Always include +C when finding an indefinite integral.
All right guys, Mr. Antonucci here, and we’re getting into Section 5 of Chapter 6. In this section we’re going to talk about the indefinite integral, and later we’ll get into the method of substitution. We’ll save substitution for a later video. In this one, we’re focusing on how to find indefinite integrals.
The Fundamental Theorem of Calculus establishes an important relationship between definite integrals and antiderivatives. A definite integral can be found easily if an antiderivative of the function can be found. Because of that, it’s customary to use the integral symbol as an instruction to find all antiderivatives of a function.
You can think of the integral symbol as an operation, kind of like a square root symbol or a multiplication symbol. It tells you to take the antiderivative.
The integral of f(x) dx is called the indefinite integral because there are no limits on the integral symbol. It is defined as the antiderivative plus C, where F is any function whose derivative is f, and C is a number called the constant of integration.
Let’s look at something important here. The derivative of x^2 + 5 is 2x.
The derivative of x^2 + 10 is also 2x.
The derivative of x^2 + 15 is still 2x.
In fact, the derivative with respect to x of x^2 plus any constant is equal to 2x. Because of that, the indefinite integral of 2x dx must include + C.
When we find an indefinite integral, we always write + C to account for that unknown constant. The antiderivative is actually a family of functions that all share the same derivative.
Let’s look at a quick example.
Find the integral of x^2 + 1 dx.
The antiderivative of x^2 is x^3 / 3.
The antiderivative of 1 is x.
So the result is:
x^3 / 3 + x + C
You can always check your answer by taking the derivative.
The derivative of x^3 / 3 is x^2, the derivative of x is 1, and the derivative of a constant is 0. That gets us back to the original integrand.
The process of finding either an indefinite integral or a definite integral is called integration.
The function inside the integral symbol, between the integral sign and the dx, is called the integrand.
Here’s an important distinction.
A definite integral has limits, and its answer is a number.
For example, the definite integral from 0 to 2 of x^2 dx is the antiderivative x^3 / 3 evaluated at 2 and 0. Plugging in 2 gives 8/3, plugging in 0 gives 0, so the result is 8/3.
An indefinite integral has no limits. You take the antiderivative and add + C. The answer is a family of functions whose common derivative is the integrand.
Here is a table of basic functions and their antiderivatives. If you see an integral with no number written, it’s understood to be a 1.
The integral of 1 dx is x + C.
For trigonometric functions, notice that we don’t integrate sec(x) directly. We integrate sec^2(x), because sec^2(x) is the derivative of tan(x). So the antiderivative of sec^2(x) is tan(x).
Similarly:
The antiderivative of sec(x)tan(x) is sec(x)
The antiderivative of csc(x)cot(x) is -csc(x)
The antiderivative of csc^2(x) is -cot(x)
For inverse trig functions:
Since the derivative of arcsin(x) is 1 / sqrt(1 - x^2), the antiderivative of 1 / sqrt(1 - x^2) is arcsin(x)
The antiderivative of 1 / (1 + x^2) is arctan(x)
The integral of e^x is e^x + C.
The integral of 1/x is ln|x| + C.
When you integrate a^x, where a is positive and not equal to 1, the result is a^x divided by ln(a), plus C.
Part A:
The integral of x^4 dx is x^5 / 5 + C.
Part B:
The integral of sqrt(x) dx can be rewritten as x^(1/2) dx.
Add one to the exponent to get 3/2, multiply by the reciprocal, and add + C.
Part C is a little trickier. There’s no direct rule for the integral of sin(x) / cos^2(x), so we rewrite it.
sin(x) / cos^2(x can be written as:
sin(x)/cos(x) * 1/cos(x)
That simplifies to:
tan(x) * sec(x)
Now we haven’t done any calculus yet — we just rewrote the integrand. The antiderivative of sec(x)tan(x) is sec(x), so the result is:
sec(x) + C
The integral of 1/x dx is ln|x| + C.
The integral of 1/x^2 dx can be rewritten as x^(-2).
Add one to the exponent and multiply by the reciprocal to get:
-1/x + C
The integral of 1/x^3 dx is x^(-3).
Add one to the exponent to get -2, multiply by the reciprocal, and simplify to:
-1/(2x^2) + C
If you have the integral of x * sqrt(x) dx, rewrite sqrt(x) as x^(1/2). Since x has an exponent of 1, when you multiply you add the exponents to get x^(3/2).
Now apply the power rule:
Add one to get 5/2
Multiply by the reciprocal
The result is:
(2/5)x^(5/2) + C
You could also rewrite this as:
(2/5)sqrt(x^5) + C
Both forms are equivalent, and on an AP exam you may see either one.
That’s it for this one. Reach out if you have any questions.
Up Next: Properties of Definite Integrals