When none of the techniques you’ve learnt so far (u-substitution 1, 2,3,4 and integration by parts seem to work, you might consider trigonometric substitution. Basically, if you have a function that reminds you of a trigonometric identity, you let that function be equal to the result of the identity, for example cos²x from 1-sin²x, and then differentiate. To make things more clear I’m going to use this example: U substitution and integration by parts are not going to be very helpful. What we are gonna do instead is let x=sin(theta)… Read more Integration techniques | Trigonometric substitution →
You might think: what the hell are you talking about? I’m talking about this: Looks crazy, right? After this lesson it’s gonna look cool, trust me! Let’s calculate the integral of x with respect to x². One day I thought: “what is the integral of a function of x with respect to x²? That’s impossible I guess.” A couple of months later, that is less than a week ago, I realised: “substitution!”. If we know how to integrate with respect to x, u, t or whatever letter it is, then… Read more Integration with respect to a function of x →
U-substitution is the most common technique used in integration. It can happen, however, that it doesn’t work out, no matter what you try to substitute. You may think of using another technique, like integration by parts, but it’s not always necessary. You realise u-sub is not working when you still see one or more x’s. For example: As you can see, we have an x that we don’t want. At this point you may think about integration by parts, and yes, that’s actually the way to go. The result is… Read more Integration techniques | U substitution – part 2 →
In this post you will learn how to produce pure copper (II) chloride (CuCl₂) using basic copper carbonate, Cu2(OH)2CO3 , which I’ll tell you how to prepare as well. These two are the kind of reactions that will result in quantitative yields if done correctly without much pouring from one container to another. Chemicals needed for 126,80 g of copper (II) chloride: 185,71 g copper sulfate pentahydrate (CuSO4 ⋅ 5H2O ) 130 g sodium bicarbonate (NaHCO3) x g hydrochloric acid (HCl), where x is equal to 5500/conc. % Prepare the… Read more How to make copper (II) chloride: base – acid method →
Chemistry happens everywhere at any time: in our body chemical reactions allow to digest and break down fats, proteins, to transport hormones, to remove old cells, to carry the oxygen we breathe to our blood; combustion is a chemical reaction, photosynthesis is a chemical reaction as well. Chemistry is a fundamental part of science without which we could not explain many, and I say many things. As a kid, you probably did the “volcano experiment” at least once: you simulated an eruption by mixing vinegar and baking soda. What happened?… Read more How to write a chemical equation | Basics →
Here you will see how to solve these integrals https://atomic-temporary-167386734.wpcomstaging.com/integration-by-parts-exercises/ and check if you got them right! Wait… we have the integral of sin(x)cos(x) again! So this means that and Easy, right? Let’s solve this integral by performing another integration by parts. so we have We can do the same thing we did for the integral of sin(x)cos(x). Therefore, We now know that this is equal to which is equal to The answer is Same thing here; And that’s it! Stay tuned for more integrals. You might find these links… Read more Integration by parts | Solution with procedure →
Practice, practice, practice. Theory is not everything. You need to apply what you learn from reading. These exercises will help you improve your “integration skills” by putting into practice what you’ve leant here https://atomic-temporary-167386734.wpcomstaging.com/integration-techniques-integration-by-parts/ about integration by parts. If you don’t remember the basic integrals and their rules, click on these two links: https://addjustabitofpi.wordpress.com/2019/10/03/fundamental-integrals/ https://addjustabitofpi.wordpress.com/2019/10/03/rules-and-properties-of-integrals/. Check your results here: https://addjustabitofpi.wordpress.com/2019/10/09/integration-by-parts-solution-with-procedure/.
Here are the solutions with procedure of these integrals: https://atomic-temporary-167386734.wpcomstaging.com/u-substitution-exercises/. If you don’t remember the rules and properties of integrals, check this out: https://atomic-temporary-167386734.wpcomstaging.com/rules-and-properties-of-integrals/.
Add just a bit of pi 