Advanced Math Solutions – Integral Calculator, the complete guide

We’ve covered quite a few integration techniques, some are straightforward, some are more challenging, but finding the “right” technique for a given function can be difficult, it requires a strategy.  Let’s start by sorting out the different techniques.

First, start by simplifying the integrand as much as possible (using simple algebraic manipulations or basic trigonometric identities).   Not necessarily a simpler form but more a form that we know how to integrate.  Go over the reviews of basic integration techniques and common functions: The basics , Common functions.


Next, look for obvious substitution (a function whose derivative also occurs), one that will get you an integral that is easy to do.  Here’s a review of simple substitution:   Substitution


If you can’t solve the integral using simplification or substitution, it’s time to step up.  Try to classify the integrand into one of the following:  product of trig powers, rational functions, radicals, or a product of a polynomial and a transcendental function.

Trig functions (product of powers of trig functions:  Trig functions,   Advanced trig functions

Rational functions ( use partial fractions if the degree of the numerator is less than the degree of the denominator, otherwise use long division), here’s how it’s done: Partial fractions ,  Long division

Integration by parts (product of a polynomial and a transcendental function: Integration by parts 

Radicals (use trig substitution if the integral contains sqrt(a^2+x^2) or sqrt(x^2-a^2), for (ax+b)^1/n try simple substitution ): Trig substitution ,  Advanced trig substitution


If none of the above techniques work, you should take some more aggressive measures; advanced algebraic manipulations, trig identities, integration by parts with no product (assume 1 as a multiplier). In some cases you need to use multiple techniques.  Here are some examples of advanced integration: Advanced integration 

The best strategy is to assume easy until easy doesn’t work, always try the simplest techniques first, and remember there is more than one way to solve an integral.

Cheers,
Michal

5 Things you need to know to help your kids succeed in Math

Your child wants to learn, but she wants to learn her way.  She wants to understand the material covered in class.  She wants to be able to look up answers just as she does for any other subject (looking up answers in Wikipedia is not cheating its learning, so does looking up answers to solve math problems).   Most importantly she can, she can learn math, there’s no such thing as "math genes".

Do not use shortcuts.  Some teachers teach shortcuts to solve math problems (mainly b/c of the misconception that kids can't or don’t want to learn math).  But shortcuts make math a matter of  rote manipulation of formulas and require your child to memorize tons of rules and formulas.  Shortcuts are great only after your child understands the topic, concept, methods and techniques.
Math is about solving problems not following rules!

Math is all about building blocks.  Your child needs a good math foundation on which she keeps adding layers.  She can't skip one part and still be able to do the rest. Later material depends on earlier material .  No matter what the reason is (can be due to poor teaching, poor experience, or simply need for a refresher), your child must fill in the blanks and keep up with class.
Symbolab introduces “interim steps” to help with earlier material that your child may not yet understands completely.  For example, to solve logarithmic equations your child must have solid knowledge of both simple equations and logarithms. Here’s how it’s done.

Keep it simple! Most math problems require some kind of simplification.  The best strategy for solving a big problem is to break it into smaller, more familiar problems.  Use the same strategy to learn math, keep it simple.

Practice, practice and practice, learning math takes practice, lots of practice.  Just like running, it takes practice and dedication.    The more she practices, the less likely your child will feel completely lost. why?  Because through practice she will have seen it all.   To help her practice start here, take a look at the examples, type in any problem and carefully review the steps.   You can add notes, rate, save to your virtual notebook; build her practice zone.

All it takes is hard work, confidence and the right tools, she can do it, we are here to help.

Cheers,
Michal

Advanced Math Solutions – Integral Calculator, integration by parts, Part II

In the previous post we covered integration by parts.  Quick review:
Integration by parts is essentially the reverse of the product rule. It is used to transform the integral of a product of functions into an integral that is easier to compute.

Integration by parts formula: \int\:uv'=uv-\int\:u'v

In this post we’ll cover some more advanced examples.

Let’s start with examples using both integration by parts and substitution (click here):

Here’s another example using integration by parts and substitution (click here):

Don’t forget to add a constant

Here’s a tricky one, using integration by parts twice and some manipulation (click here):

Here’s another tricky example (click here):

If you've gotten this far, you sure know how to integrate.  Good job!

Cheers,
Michal