In this video, I break down the step-by-step process of integrating the rational function (1 - x)/x³ with respect to x. This type of question is common in calculus problems for students preparing for exams like WAEC, JAMB, NECO, and university-level math. Whether you are a secondary school student, an undergraduate, or someone brushing up on integration skills, this video gives you a clear and simple guide to solving this integral.
Integration is one of the most important parts of calculus, and it helps us to find areas under curves, solve differential equations, compute accumulated quantities, and so on. The particular integral we are solving in this video is:
∫ (1 - x)/x³ dx
This looks a bit complex at first sight, but the trick is to simplify the expression before integrating. I show how to split the function into simpler parts, and then apply the standard rules of integration to find the final answer. The beauty of this approach is that it helps you understand how algebra and calculus come together in integration.
Rewrite the Function
We start by rewriting the integrand (1 - x)/x³ as a sum of simpler terms that are easier to integrate individually. The idea is to separate the numerator:
(1 - x)/x³ = 1/x³ - x/x³
Now simplify each term:
1/x³ stays as it is
x/x³ = 1/x²
So, we now have:
∫ (1/x³ - 1/x²) dx
This is easier to integrate term by term.
Apply the Power Rule of Integration
The power rule of integration says:
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C, where n ≠ -1
Let’s apply it to each term:
∫ x⁻³ dx = x⁻² / -2 = -1/(2x²)
∫ x⁻² dx = x⁻¹ / -1 = -1/x
So, the solution becomes:
∫ (1/x³ - 1/x²) dx = -1/(2x²) + 1/x + C
Where C is the constant of integration.
This is the final answer. In the video, I explain this in detail using clear handwriting and calm voice, breaking it into bits so that even if you are seeing integration for the first time, you will be able to understand and follow along. I also take time to explain why we separate the fraction and how each part fits into the general rule for integration of powers.
Why You Need to Master This
This kind of question is very common in exams. In JAMB or WAEC, they usually come as part of multiple-choice questions or even structured long questions. Understanding this method will not only help you solve this particular question, but also help you solve other integrals involving algebraic fractions, especially when the numerator is a polynomial of lower degree than the denominator.
Practice More
Try solving this similar integral on your own:
∫ (2 - 3x)/x⁴ dx
Follow the same steps and see if you can break it down and integrate each term. If you like, drop your answer in the comment section of the video and I will review it for you.
You can also try the integral:
∫ (x² - 1)/x⁴ dx
Let’s see how far you can go with it.
On this channel, I specialize in breaking down mathematics and science concepts in simple language. Whether it’s calculus, algebra, trigonometry, or even basic arithmetic, I teach in a way that everybody can understand. I also do videos in Pidgin English for those who prefer to learn in a more relaxed style.
Make sure you go through my other videos on integration, differentiation, and algebra. I’ve covered topics like definite integrals, substitution method, integration by parts, and solving area under curves. Each video is short and straight to the point, with examples and exercises for you to try.
If this video helped you understand how to integrate a function like (1 - x)/x³, then please help others too by sharing the video on your WhatsApp group, Facebook, and other social media platforms. Someone might be struggling with this topic and your share might be the help they need.
Also, don’t forget to subscribe to the channel if you haven’t already. Subscribing helps the channel grow and also makes sure you don’t miss future videos when they drop. Click the like button too if you learned something new today.
You can also leave a comment to tell me what other math or science topics you would like me to cover. I always read comments and I try to respond to every request.
Let’s keep learning and growing together. Maths no hard if dem break am down well. I dey here to help you break am down step by step.
Thanks for watching, and see you in the next video.
### Hashtags for Easy Search
#IntegrationMadeEasy
#CalculusTutorial
#IndefiniteIntegrals
#JAMB2025
#WAEC2025
#LearnMathOnline
#IntegrationStepByStep
#NigerianMathTutor
#MathWithInstructorAlison
#SimpleIntegration
#MathsForJamb
#InstructorAlisonExplains
#ExamSuccess
#MathsInPidgin
#AlgebraAndCalculus
#IntegrationWithoutCalculator
#SecondarySchoolMath
#MathematicsTutorial
#WAECMaths
#JAMBMathTips