Advanced Math Solutions – Integral Calculator, common functions

In the previous post we covered the basic integration rules (click here).  Before we continue with more advanced techniques, we will cover some common integrals (reciprocal, exponential and trigonometric functions).  You will be using the common integrals a lot, so get to know them well.

Reciprocal & Exponential:
\int\frac{1}{x}dx=\ln|x|+C
\int e^x dx=e^x+C
\int a^x dx=\frac{a^x}{\ln(a)}+C

Trigonometry:
\int\cos(x)dx=\sin(x)+C
\int\sin(x)dx=-\cos(x)+C
\int\sec^2(x)dx=\tan(x)+C
\int\csc^2(x)dx=-\cot(x)+C

The remaining trig functions can be integrated using advanced techniques, but that’s for later.


Let’s take a look at a few examples (we’re doing it slowly)
Starting with basic trigonometric functions (click here):

Exponential function (click here):

Here’s an example using the sum rule, power rule, and common integrals (click here):

Still doing good.

Cheers,
Michal

Advanced Math Solutions – Integral Calculator, the basics

Integration is the inverse of differentiation.   Even though derivatives are fairly straight forward, integrals are not. Some integration problems require techniques such as substitution, integration by parts, trigonometric substitutions, or possibly more than one method.   We will walk you through slowly, starting with the basic integration rules:  the constant multiplication rule, the power rule, and the sum rule.

Some common functions you should get familiar with (we’ll show you more later):
\int a dx = ax + C
\int x dx = \frac{x^2}{2} + C

One more thing to remember, always add the constant of integration C.

Let’s start with the Power Rule: \int x^n dx = \frac{x^{n+1}}{n+1} + C,\quad n\ne-1
The power rule simply tells you to divide by n+1 (the power + 1) and increase the power by 1, it’s that simple.  Here’s an example of how it works (click here):

Let’s continue with the constant multiplication rule (click here):
\int af(x) dx = a\int f(x)dx

The constant multiplication rule simply tells to take out the constant

Moving on to the Sum Rule (click here):

Putting it all together (click here):

That wasn’t too bad.  If you’d like to take a pick at some more advanced integrals click here

Cheers,
Michal

Lies, Damned Lies, and Statistics

Statistics is about analyzing data, for instance the mean is commonly used to measure the “central tendency” of the data set (simply the average), or the median that measures the true middle of the data set.  Other basic methods that are fairly intuitive are the mode that finds the value that occurs most often, and the range that measures the difference between the largest and smallest numbers.  Some of the advanced methods that require more complex calculations are the variance, and standard deviation that measure how spread out the numbers are (measure of dispersion).

Why do we need all this?  To be able make decisions or even predictions based on a sample data set alone.  Symbolab now implements quite a few statistics methods to help you sort out the data step by step (No decisions, predications or “lies” just yet…).

Starting with the basics, here’s an example of the mean (click here)

here’s an example of the median (click here)

Let’s step up, here’s an example of the variance (click here)

Standard deviation, piece of cake (click here)

Click here to check the different statistics methods and examples

Cheers,
Michal