The Indefinite Integral and the Net Change

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Indefinite Integrals and Antiderivatives

In the previous module, we discussed the difference between a

definite integral, e.g.

$\displaystyle\int_a^b f(x)\,dx$ , an

integral function, e.g.

$\displaystyle\int_a^x f(t)\,dt$ . We now introduce notation for an antiderivative, called an

indefinite integral, e.g.

$\displaystyle\int f(x)\, dx$ . In other words, If

$F'(x)=f(x)$ , we say

$\displaystyle F(x)=\int f(x)\, dx$ .

By the Fundamental Theorem of Calculus I, since

$\qquad F'(x)=\frac{d}{dx}\int_a^x f(t)\, dt=f(x), \qquad\text{ we have }\qquad \int f(x)\, dx = F(x) = \int_a^x f(t)\, dt.$

This shows that the integral function is indeed an antiderivative; it is the antiderivative of its integrand. All antiderivatives are the same, up to adding a constant, and indeed changing the value of $a$ in $F(x)=\int_a^x f(t)\, dt$ changes $F(x)$ by a constant, so both antiderivatives and integral functions are only defined up to a constant.

Don't get too bogged down in the previous two paragraphs, but you do need to understand the three types of "integrals" listed above. From now on, the notation $\int f(x)\, dx$ will refer to the antiderivative, and we'll usually just call it "the antiderivative of $f(x)$ " or "the integral of $f(x)$ " (with respect to $x$ ) for short.

$\displaystyle\int f(x)\, dx$ is the antiderivative of

$f$ .

Notice the difference between a definite integral and an indefinite integral. With our new notation, the Fundamental Theorem of Calculus II says that

$\int_a^b f(x)\,dx=\int f(x)\,dx\left |\begin{array}{c} ^b \\ _a \end{array}\right .$

This video gives the basic techniques for finding antiderivatives.

The Fundamental Theorem of Calculus

The Indefinite Integral and the Net Change

Substitution

Area Between Curves

Volumes

Integration by Parts

Integrals of Trig Functions

Trig Substitutions

Partial Fractions

Strategies of Integration

Improper Integrals

Differential Equations

Models of Growth

Infinite Sequences

Infinite Series

Integral Test

Comparison Tests

Convergence of Series with Negative Terms

The Ratio and Root Tests

Strategies for testing Series

Power Series