We could calculate the function at a few points and add up slices of width Δx like this (but the answer won't be very accurate):

We can make Δx a lot smaller and add up many small slices (answer is getting better):

And as the slices approach zero in width, the answer approaches the true answer.

We now write dx to mean the Δx slices are approaching zero in width.

The symbol for 'Integral' is a stylish 'S'
(for 'Sum', the idea of summing slices):

So Integral and Derivative are opposites.

The integral of the flow rate 2x tells us the volume of water:

And the slope of the volume increase x²+C gives us back the flow rate:

And hey, we even get a nice explanation of that 'C' value ... maybe the tank already has water in it!

• The flow still increases the volume by the same amount
• And the increase in volume can give us back the flow rate.

Which teaches us to always add '+ C'.

Other functions

Well, we have played with y=2x enough now, so how do we integrate other functions?

If we are lucky enough to find the function on the result side of a derivative, then (knowing that derivatives and integrals are opposites) we have an answer. But remember to add C.

Example: what is ∫cos(x) dx ?

From the Rules of Derivatives table we see the derivative of sin(x) is cos(x) so:

∫cos(x) dx = sin(x) + C

But a lot of this 'reversing' has already been done (see Rules of Integration).

Example: What is ∫x³ dx ?

On Rules of Integration there is a 'Power Rule' that says:

∫xⁿ dx = xⁿ⁺¹n+1 + C

We can use that rule with n=3:

∫x³ dx = x⁴4 + C

Knowing how to use those rules is the key to being good at Integration.

So get to know those rules and get lots of practice.

Learn the Rules of Integration and Practice! Practice! Practice!
(there are some questions below to get you started)

Definite vs Indefinite Integrals

6 Basic Integration & Applicationsap Calculus 14th Edition

We have been doing Indefinite Integrals so far.

A Definite Integral has actual values to calculate between (they are put at the bottom and top of the 'S'):

Read Definite Integrals to learn more.

1. The Indefinite Integral

Recall that if I is any interval and f is a function definedon I, then a function F on I is anantiderivative of f if F'(x) = f(x) for all x ∈ I. A corollary ofthe Mean Value Theorem tells us that if G is any other antiderivative of f on I, then there is a constant C such that G(x) = F(x) + C.

Definition

Let I be an interval and f a function defined on I.Then the indefinite integral of f is defined tobe the set of all antiderivatives of f on I. We denote this set by:

We call f the integrand and x the variable ofintegration.

Since all the antiderivatives of f differ only by a constant, it is customary to write:

where F is any particular antiderivative, and C is the constant of integration which implicitly takes on all real number values.

2. The Definite Integral

Definition

Let f be a function defined on a closed interval [a, b]. Given any n we let Δx=(b-a)/n, and we let x_k = a + k Δx. For each k, select sample pointsx_k* ∈[x_k-1, x_k]. Wedefine the definite integral of f on [a, b]to be

provided this limit exists and does not depend on the choice of sample points x_k*.

If the limit depends upon the choice of sample points, or is undefined forany choice of sample points, then the integral does not exist.

We call a the lower limit of theintegral, and b the upper limit. We callf the integrand.

We also make the defintions:

The integral sign '∫' is an archaic 'S' and stands for 'sum' (as does ∑, of course).

If it exists, the definite integral gives you a number as its result. In this regard, x is a 'dummy variable' and the dx reminds us what variable we are integrating with respect to. We can just as easily replace x with t and write:

and the numerical value of the definite integral is unchanged.

Geometrically, if f(x) ≥ 0 we interpret the integral as thearea between the graph y=f(x), the x-axis, andthe lines x=a and x=b. More generally,we think of the integral as a 'signed' area, where the integral is equal tothe area above the x-axis less the area below the x-axis.

The sum:

in the definition of the definite integral is called a Riemann sum; the definite integral is sometimes called a Riemann integral (to distinguish it from other more general integrals used by mathematicians). Whenever you have a limit as n goes to infinity of a Riemman sum, you can replace itby an appropriate integral. This observation is critical in applicationsof integration.

For certain simple functions, you can calculate an integral directly usingthis definition. However, in general, you will want to use the fundamentaltheorem of calculus and the algebraic properties of integrals.

3. Rules of Integration

Let f and g be functions and let a, b and c be constants, and assume that for each fact all the indicated definite integrals exist. Then the following are true:

1. Constants can be pulled out of integrals:

2. The integral of the sum of two functions equals the sum of the integrals of each function:

3. The integral of the difference of two functions equals the difference of the integrals of each function:

4. The integral from a to b of a function equals the integral from a to c plus the integral from c to b:

6 Basic Integration & Applicationsap Calculus Solver

Note that there are no general rules for integrals of products andquotients. Such integrals can sometimes, but not always, be calculated usingsubstitution or integration by parts.

6 Basic Integration & Applicationsap Calculus 2nd Edition

We can also give some facts about inequalities involving definite integrals.

1. If f(x) ≥ 0 on [a, b] then

2. If f(x) ≤ g(x) on [a, b] then

3. The absolute value of an integral is less than the integral of the absolute value.

4. Basic Integrals

Integrals of Powers

Integrals of Exponentials

Integrals of Trigonometric Functions

Integrals Involving Inverse Trigonometric Functions