# I'm looking for examples of integration that use this method

• Ryuk1990
In summary, my professor is teaching us to use integration by finding the derivative of a function. I think he calls this technique the product rule or reverse chain rule. He uses u-substitution to show how to do more complicated examples. Most calculus tutorial sites use u-subs or integration by parts to show how to do the more complicated examples.
Ryuk1990
My professor is teaching us to use integration by finding the derivative of a function. I think he calls this technique the product rule or reverse chain rule.

For example, integrate sin(4x) dx. I find the derivative of the inner function which is 4 and I even it out by multiplying the whole package by 1/4. The integral of sin is -cos so it's -1/4cos4x.

This is a very simple example and I'm looking for websites that show much harder examples that use this technique. The problem is that most calculus tutorial sites seem to use u-subs or integration by parts to show how to do the more complicated examples.

not sure what you want here. Why not try using integraption by parts to derive the Taylor series for a function.

Ryuk1990 said:
My professor is teaching us to use integration by finding the derivative of a function. I think he calls this technique the product rule or reverse chain rule.
You are confusing two different techiques here. Integration by parts is the reverse of the product rule in differentiation. Integration by substitution is the reverse of the chain rule in differentiation.
Ryuk1990 said:
For example, integrate sin(4x) dx. I find the derivative of the inner function which is 4 and I even it out by multiplying the whole package by 1/4. The integral of sin is -cos so it's -1/4cos4x.

This is a very simple example and I'm looking for websites that show much harder examples that use this technique. The problem is that most calculus tutorial sites seem to use u-subs or integration by parts to show how to do the more complicated examples.
Your calculus textbook should have numerous examples of ordinary substitution (what you are calling u-sub).

Ryuk1990 said:
My professor is teaching us to use integration by finding the derivative of a function. I think he calls this technique the product rule or reverse chain rule.

For example, integrate sin(4x) dx. I find the derivative of the inner function which is 4 and I even it out by multiplying the whole package by 1/4. The integral of sin is -cos so it's -1/4cos4x.

This is a very simple example and I'm looking for websites that show much harder examples that use this technique. The problem is that most calculus tutorial sites seem to use u-subs or integration by parts to show how to do the more complicated examples.

What you are describing IS a u-substitution.

sin(4x)dx

let u=4x which means du = 4dx => 1/4du = dx

Ah well I suppose it's the notation of the u-subbing that confuses me. I like the intuitive shortcut of the method which is what my teacher shows us.

Thats something most people can develop after doing LOTS of u-substitution problems. I know I did. After a while you just see it and your u-substituting becomes an 'intuitive shortcut method'. If you want to understand it, learn both ways.

## 1. What is integration and why is it important?

Integration is a mathematical concept that involves finding the area under a curve or the sum of infinitesimal quantities. It is important because it allows us to solve problems involving continuous quantities, such as velocity, acceleration, and volume.

## 2. What are some common methods of integration?

Some common methods of integration include the use of basic integrals, substitution, integration by parts, trigonometric substitution, and partial fractions. Each method has its own advantages and is used to solve different types of problems.

## 3. How do you use substitution as a method of integration?

Substitution involves replacing a variable in an integral with a new variable in order to simplify the integral. This method is useful when the integrand contains a complicated function or when the integral resembles a known integral. The substitution should be chosen carefully to make the integral easier to solve.

## 4. What is integration by parts and when is it used?

Integration by parts is a method where the integral of a product of two functions is reduced to an integral of a simpler form. It is used when the integral contains a product of two functions or when the integral resembles the product rule of differentiation. This method is helpful in solving integrals involving trigonometric functions, logarithmic functions, and exponential functions.

## 5. Can you provide an example of integration using the method of partial fractions?

Yes, an example of integration using partial fractions is the integral of 1/(x^2+2x+3). This integral can be rewritten as 1/((x+1)^2+2), and by using partial fractions, it can be solved as a sum of simpler integrals 1/((x+1)^2+2) = A/(x+1) + B/((x+1)^2+2), where A and B are constants. This method is useful for solving integrals involving rational functions with complex denominators.

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