Khan Academy on a Stick
Indefinite and definite integrals
Indefinite integral as antiderivative. Definite integral as area under a curve. Integration by parts. Usubstitution. Trig substitution.
Indefinite integral as antiderivative
You are very familiar with taking the derivative of a function. Now we are going to go the other way aroundif I give you a derivative of a function, can you come up with a possible original function. In other words, we'll be taking the antiderivative!
Riemann sums and definite integration
In this tutorial, we'll think about how we can find the area under a curve. We'll first approximate this with rectangles (and trapezoids)generally called Riemann sums. We'll then think about find the exact area by having the number of rectangles approach infinity (they'll have infinitesimal widths) which we'll use the definite integral to denote.
Integration by parts
When we wanted to take the derivative of f(x)g(x) in differential calculus, we used the product rule. In this tutorial, we use the product rule to derive a powerful way to take the antiderivative of a class of functionsintegration by parts.

Usubstitution
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Using Usubstitution to find the antiderivative of a function. Seeing that Usubstitution is the inverse of the chain rule.

Usubstitution example 2
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Another example of using Usubsitution

Usubstitution Example 3
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Manipulating the expression to make usubstitution a little more obvious.

Usubstitution with ln(x)
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Doing usubstitution with ln(x)

Doing usubstitution twice (second time with w)
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Example where we do substitution twice to get the integral into a reasonable form

Usubstitution and back substitution
Using usubstitution and "back substituting" for x to simplify an expression

Usubstitution with definite integral
Example of using usubstitution to evaluate a definite integral

(2^ln x)/x Antiderivative Example
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Finding ∫(2^ln x)/x dx

Another usubstitution example
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Finding the antiderivative using usubstitution.
Usubstitution
Usubstitution is a musthave tool for any integrating arsenal (tools aren't normally put in arsenals, but that sounds better than toolkit). It is essentially the reverise chain rule. Usubstitution is very useful for any integral where an expression is of the form g(f(x))f'(x)(and a few other cases). Over time, you'll be able to do these in your head without necessarily even explicitly substituting. Why the letter "u"? Well, it could have been anything, but this is the convention. I guess why not the letter "u" :)
 Riemann sums and integrals
 Intuition for Second Fundamental Theorem of Calculus

Evaluating simple definite integral
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 Definite integrals and negative area
 Area between curves
 Area between curves with multiple boundaries

Challenging definite integration
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2010 IIT JEE Paper 1 Problem 52 Periodic Definite Integral. The second term at about minute 14 should have a positive sign. Luckily, it doesn't effect the final answer!

Introduction to definite integrals
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Using the definite integral to solve for the area under a curve. Intuition on why the antiderivative is the same thing as the area under a curve.

Definite integrals (part II)
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More on why the antiderivative and the area under a curve are essentially the same thing.

Definite Integrals (area under a curve) (part III)
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More on why the antiderivative and the area under a curve are essentially the same thing.

Definite Integrals (part 4)
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Examples of using definite integrals to find the area under a curve

Definite Integrals (part 5)
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More examples of using definite integrals to calculate the area between curves

Definite integral with substitution
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Solving a definite integral with substitution (or the reverse chain rule)
Definite integrals
Until now, we have been viewing integrals as antiderivatives. Now we explore them as the area under a curve between two boundaries (we will now construct definite integrals by defining the boundaries). This is the real meat of integral calculus!

Introduction to trig substitution
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 Another substitution with x=sin (theta)

Integrals: Trig Substitution 1
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Example of using trig substitution to solve an indefinite integral
 Trig and U substitution together (part 1)
 Trig and U substitution together (part 2)
 Trig substitution with tangent

Integrals: Trig Substitution 2
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Another example of finding an antiderivative using trigonometric substitution

Integrals: Trig Substitution 3 (long problem)
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Example using trig substitution (and trig identities) to solve an integral.
Trigonometric substitution
We will now do another substitution technique (the other was usubstitution) where we substitute variables with trig functions. This allows us to leverage some trigonometric identities to simplify the expression into one that it is easier to take the antiderivative of.
Fundamental Theorem of Calculus
You get the general idea that taking a definite integral of a function is related to evaluating the antiderivative, but where did this connection come from. This tutorial focuses on the fundamental theorem of calculus which ties the ideas of integration and differentiation together. We'll explain what it is, give a proof and then show examples of taking derivatives of integrals where the Fundamental Theorem is directly applicable.