How Can I Master Integration Techniques in Calculus?

[sarah2529]

I'm studying Integral Calculus on my own with the help of a book and videos around the net. I found this site http://justmathtutoring.com/ and notice that he use only some basic samples. Most of the question I found on my book uses only basic questions.

I'm wondering what if the problem is to integrate( (x-1)/4 ) or maybe integrate( ((x-2)(x-1))/(x-10) )... Differentiation is quite easy because there's what we call quotient rule but with integrals there are none.

So my question is, how can I integrate this? Am I in the lost track or maybe I really didn't understand the topic?

Another question by the way is with this bunch of ways to get the integration(ie. by substitution, by parts and etc). How do I know which one to use?\

Thank you very much
Sarah2529

 

[LCKurtz]

I'm afraid the only way you will know which method to use is to use the experience you gain by practicing. There are some things to look for such as looking for the du in a u substitution. For example if you want to integrate (x^3 + 5)^4 (x^2), the fact that the x^2 term is there suggests u = x^3 + 5 might work. Some, like your (x-1)/4 would just be written as (1/4)x - 1/4 and worked directly. For one like (x-2)(x-1))/(x-10) I would multiply the numerator out and do long division getting a polynomial and a remainder of the form A/(x-10) which can be done by substitution. But, again, to answer your question, there isn't any way I know of except practicing to gain experience.

 

[statdad]

Once you move past the process of integrating simple polynomials, things can get quite involved quickly.

Your first question is easily handled, because the denominator is a constant: just remember that "constants can be factored through the integral", which is typically written as this "integration rule"

<br /> \int c f(x) \, dx = c \int f(x) \, dx<br />

For problems of your second type (ratio of polynomials, numerator higher degree than denominator), the general procedure is to write the integrand as a sum of "proper" fractions. The first step requires the use of partial fractions. A fairly straightforward introduction can be found at

http://www.intmath.com/Methods-integration/11_Integration-partial-fractions.php

You can find discussion of other integration methods at that site. However, there is no
universal method - nothing works on every integral that one encounters, and for more complicated integration problems one must know a great deal of math past calculus before the problem can be identified, let alone solved.

Good luck with your studies.

 

[wofsy]

I'm studying Integral Calculus on my own with the help of a book and videos around the net. I found this site http://justmathtutoring.com/ and notice that he use only some basic samples. Most of the question I found on my book uses only basic questions.

I'm wondering what if the problem is to integrate( (x-1)/4 ) or maybe integrate( ((x-2)(x-1))/(x-10) )... Differentiation is quite easy because there's what we call quotient rule but with integrals there are none.

So my question is, how can I integrate this? Am I in the lost track or maybe I really didn't understand the topic?

Another question by the way is with this bunch of ways to get the integration(ie. by substitution, by parts and etc). How do I know which one to use?\

Thank you very much
Sarah2529

There is no set of integration techniques that works for all problems. Much of mathematics is the search for ways to integrate functions.

For beginning calculus of one variable you should know how to

Reverse the Chain Rule
Integrate by parts
The anti-derivatives of polynomials, trig functions, the logarithm and exponential

You should understand the fundamental theorem of calculus.

 

[sarah2529]

Ohhh I replied so late. haha! Anyway, thanks for the help everyone. I understand most of the topics in integration. Thank you very much on my book, justmathtutoring and also MIT!

I guess I really need to analyze the problem very well then apply the theorems.

Is there any site out there that has bunch of complicated examples for integration? I want to practice very well so I can prove to myself that I really understand it. I want to get an A++++++ on my calculus class. Yiippeee! haha! (^_^)

I love math so much! I'm in love with it! nahhhh. hehe :P

 

[lurflurf]

A fun saturday night would be reproducing a nice integral table.
Since you have the specific goal of doing well in introduction to calculus here are some practice sheets toward that.
http://www.integral-table.com/integral-table.pdf
dartmouth math 11
dartmouth math 12
http://math.georgiasouthern.edu/~asills/teach/fall08/integration%20practice.pdf
http://www.math.montana.edu/~pernarow/M182/2007Sp/notes/Integration_Problems.pdf
cambidge
http://www.math.jmu.edu/~taal/236_2001post/236integration_practice.pdf