In this post I am going to explain step-by-step how to integrate the function xln(x). We can solve it by using integration by parts (click here if you want to… Read more Integral of xln(x) →
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 →
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 →
Here you will see how to solve these integrals https://addjustabitofpi.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://addjustabitofpi.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.com/2019/10/03/fundamental-integrals/ https://addjustabitofpi.com/2019/10/03/rules-and-properties-of-integrals/. Check your results here: https://addjustabitofpi.com/2019/10/09/integration-by-parts-solution-with-procedure/.
Here are the solutions with procedure of these integrals: https://addjustabitofpi.com/u-substitution-exercises/. If you don’t remember the rules and properties of integrals, check this out: https://addjustabitofpi.com/rules-and-properties-of-integrals/.
Here are some indefinite integrals you can practice with using u-sub. If you can’t solve all of them, don’t discourage. You can see the procedure for each here: https://addjustabitofpi.com/category/solutions/. This will help you understand the logics of integrals and you will eventually get the hang of it and be able to solve them right away! If you don’t remember the rules and properties of integrals, click this link: https://addjustabitofpi.com/rules-and-properties-of-integrals/. Notice that cot (cotangent) is the reciprocal of tan (tangent), which means that cot(x)=cos(x)/sin(x). Remember that sec (secant) is the reciprocal… Read more U-substitution | Exercises →
U-substitution is a very useful technique for integration. You can apply it in every case, but it doesn’t always make things easier. But here comes this new technique in help!… Read more Integration techniques | Integration by parts →