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Posts Tagged ‘integrals’

Integration by Trigonometric Substitution

Thursday, September 10th, 2009

How to Solve an Integration Problem by Trigonometric Substitution

Description

This video clearly illustrates how to solve an integration problem using trigonometric substitution. One example problem is provided in the video.

Overview

Trigonometric substitution works the same way as normal substitution, only you substitute in trigonometric functions, and each trigonometric function can only be substituted for a particular pattern. These are the patterns to watch for and what you can substitute in for them:

a^2 – x^2 uses x = a * sin(theta)

a^2 + x^2 uses x = a * tan(theta)

x^2 – a^2 uses x = a * sec(theta)

All of these also have the option of including a square root with them, but it doesn’t matter – you can use the substitution without the square root. Normally after finding the x value you will take a derivative so you have the value of dx To find other values needed,SOHCAHTOA is often used with a right triangle picture. Once all of your values have been solved for, you can plug them all back into your original equation and solve.

Tags: antiderivatives, antidifferentiation, Calculus, derivatives, differentiation, integrals, integration by substitution, Math, secant, sine, tangent, trig functions, trigonometric substitution, trigonometry
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Integration by Substitution

Thursday, September 10th, 2009

How to Solve an Integration Problem by Substitution

Description

This video shows an example of how to solve the same problem with and without substitution, to express how much easier substitution is. This video provides several example problems with clear explanations and solutions for each one.

Overview

The substitution rule comes in very useful no matter how you are trying to solve an integral. When you are presented with something that looks like you may need the chain rule to solve it, or you are presented with a polynomial term in the denominator of your integral, then you can use u-substitution. It is called that because the variable u is the most common one used. Once you have chosen a value for u, find the derivative and write it as du/dx = n, with n being the number value of du/dx. Now, move the du over so that you are left with a value for dx. Now skip back to your original problem. Once you replace a term with u, you are normally left with a common integral, which is easy to solve. Multiply that by dx, just like in the problem. dx will be expressed as what you just solved for. If this is a multiplication problem, move all numbers to the outside and leave yourself with just the variables. Now you should be able to solve this like a normal integration problem.

Tags: antiderivatives, antidifferentiation, Calculus, derivatives, differentiation, integrals, integration by substitution, Math, u-substitution
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Integration by Parts

Thursday, September 10th, 2009

How to Solve an Integration Problem by Parts

Description

Step-by-step tutorial of the integration by parts method. Several examples are provided in this video. Knowledge of integration by parts is required in calculus.

Overview

Very often, you will see an integral that is also a multiplication problem. There was such a thing as the product rule for derivatives, but what about for anti-derivatives? There is some sort of reverse product rule, and that is what integration by parts is. Sometimes it is also called u-substitution, but the term is misleading because substitution is something else. A basic integration by parts problem is set up like this:

$\int u\, dv=uv-\int v\, du.\!$

You are allowed to choose which value is your u and which value is your dv. Choosing involves something called LIATE – Logarithmic, Inverse Trigonometric, Algebraic, Trigonometric, Exponential. Put this on a ladder scale, with L being first and E being last. Each term you are multiplying falls on one of the parts of this chart. The highest one will be your u, and the lower one will be your dv. Sometimes you will have to perform integration by parts more than once in a problem.

Tags: antiderivatives, antidifferentiation, Calculus, derivatives, differentiation, integrals, integration by parts, LIATE, Math
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Basic Integrals

Thursday, September 10th, 2009

The Definition of an Integral

Description

This video describes what an integral or an anti-derivative is. Several examples are provided in this video. Knowledge of integrals is needed to pass calculus.

Overview

An integral is the opposite of a derivative, and therefore is often called an antiderivative. While with a differentiation problem, you need to find a derivative, if you are solving an integral you are already given a derivative and your goal is to revert it back to it\’s common form.

Example: The derivative of 2x is 2. The antiderivative or integral of 2 would be 2x.

You should already have several derivatives memorized. This makes starting to solve integrals much easier, because if you have the derivative memorixed all you need to ask yourself is \”what is that a derivative of?\” and you have your answer. For reference, the integral symbol looks like a very stretched out S symbol, which is supposed to be an italic I.

Tags: antiderivatives, antidifferentiation, Calculus, derivatives, differentiation, integrals, Math
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