Evaluating the limit of x-3 with x approaching 4 is very easy and straightforward: you just plug in the value x tends to and you’re done! Another example: How about this? As we saw here, one divided by zero equals infinity, therefore one divided by infinity equals zero. We can’t evaluate limits directly by substitution when, if we do substitute, we get an indeterminate result, like infinity/infinity, 0/0, infinity – infinity, 0 times infinity, infinity/0. When this happens, we need to rewrite the limit, without substitution, and apply L’Hôpital’s rule.… Read more L’Hôpital’s rule →

Indefinite integrals are used to find the formula for the area under a curve of function f(x), whereas definite integrals allow you to calculate the value of the area. It’s like the area of a rectangle, b*h: this is the formula, and in order to find the value you have to plug in the values of b and h. The same is for definite integrals. A definite integral is generally written like this: where a and b are called respectively lower and upper bounds. Also, As you can see, if… Read more How to solve definite integrals →

Here I talked about special functions, and at the end of the post I said I would explain how to integrate these special functions. So here I am, showing you how to integrate the functions S(x) and C(x). If you gave it a try, let me know in the comments if you used this approach! Integral of S(x) Like we did for the previous integrals, we are going to use integration by parts, as it is the only way to go. Now we can use u-substitution to solve the integral:… Read more Integral of S(x) and C(x) →

Here I talked about special functions, and at the end of the post I said I would explain how to integrate these special functions. So here I am, showing you how to integrate the functions Si(x) and Ci(x). If you gave it a try, let me know in the comments if you used this approach! Integral of Si(x) U-substitution won’t work, so let’s use integration by parts. We know that Si(x) is given by the integral of sin(x)/x, therefore the derivative of Si(x) is sin(x)/x. Integral of Ci(x) For this… Read more Integral of Si(x) and Ci(x) →

Here I talked about special functions, and at the end of the post I said I would explain how to integrate these special functions. So here I am, showing you… Read more Integral of erf(x) →

Here I talked about special functions, and at the end of the post I said I would explain how to integrate these special functions. So here I am, showing you how to integrate the function li(x). If you gave it a try, let me know in the comments if you used this approach! As we saw here, u-substitution won’t work, but integration by parts will! We know that the li(x) is the integral of 1/ln(x), so the derivative of li(x) is 1/ln(x). Now, if we use integration by parts again,… Read more Integral of li(x) →

Here I talked about special functions, and at the end of the post I said I would explain how to integrate these special functions. So here I am, showing you how to integrate the function Ei(x). If you gave it a try, let me know in the comments if you used this approach! We said that Ei(x) is equal to the integral of (e^x)/x, which means that the derivative of Ei(x) is (e^x)/x. What we need to figure out is what integration technique will actually be helpful. Honestly, u-substitution is… Read more Integral of Ei(x) →