Calculus is a part of mathematics in which you have to find out the properties of Derivatives and Integrals of a function. It is Further divided into two types, namely the differential calculus and the integral calculus. Calculus is a Nightmare to most of the students who study mathematics, and they do have a widespread question in their mind that, “Is calculus useful after my school/college life?” The answer is Yes! Calculus is used in most of the engineering streams for ‘Designing Purposes’, ‘Building Infrastructures’, in ‘Electrical Circuits ‘ and many more. There are few Integration Rules for you to learn.
- 1 Essential Integration Rules to Remember
- 2 Let’s Discuss the Methods of Integration
- 3 Conclusion
Essential Integration Rules to Remember
When you encounter integration numerical problems, you can follow the steps mentioned below to make your question a little easier.
Before you go ahead, you must know these:
Simplify the Integral as far as Possible
The most basic step of approaching any integral problem is a simplification. Simplify the given integral with the help of some basic trigonometric and algebraic identities and make an integral, easier to solve. This is one of the basic Integration Rules.
Try Simple Substitution
Sometimes, instead of going for the complex math, simple substitution does wonders and helps you to solve the given integral problem with ease. Simple substitution is vital before you opt for a proper integration method. As the name suggest this is one of the simplest Integration Rules.
Easiest Approach to Identify Integral Problems
- If the integration question is a division of polynomial by polynomial, then the partial method can be the best approach for it.
- When the integration question is a polynomial which includes exponent, logarithms, and trigonometric functions, then the integration by parts method can be the best approach for it.
- If the integration question has sin, cos and all other trigonometric terms, then maybe integration by parts will be the best way to approach it.
- When the integration question has simple roots, then substitution might be the best approach for it.
- If the integration has Complex roots with lots of variables, then Trigonometric substitution might be the best approach for it.
Try Multiple Techniques
If one technique did not do the job. Suppose the problem did not solve by substitution or simplification then you can apply partial integration or integration by parts to find the solution. Applying multiple techniques to the given integral makes the integral easy, and the complexity of it decreases.
Never Get Disappointed
Integration is not an easy thing to do. If by any chance you are not able to solve the given integral question then you have to try again with other techniques that you know. Never give up on a question just after applying a single technique and not getting the answer.
Sometimes, in integration, the types of questions are very complex and difficult. Here it requires a specific category approach for the solution. Don’t sad if you can’t solve any complex integration problems, you can always keep on trying different approaches one or the other might help you solve it. Remind yourself of the Integration Rules.
Let’s Discuss the Methods of Integration
Two of the most generally used methods to solve any integral problems are Integration by Parts and the Partial Fraction Method using the above specified Integration Rules. Although there is another way to evaluate limits but these are great as well.
Integration by Parts
Let us suppose that the question is an integral of product of two simple functions. Here, we name the first function as U and the second function as V. It is also called the UV method of integration. The formula for integration by parts is;
Here U and V, both are functions of x. For deciding which function will be the first function and which function will be the second function, we follow the ILATE approach.
- I stands for Inverse functions
- L stands for Logarithmic functions
- A stands for Algebraic functions
- T stands for Trigonometric functions
- E stands for Exponential functions
Partial Fractions Method of Integration
Suppose an integral question is in the form of division of a polynomial by a polynomial, that is P(x) / Q(x). So for the ease of solution, both numerator and denominator are approached by partial fractions at first.
Let us take an example of a non-repeated linear factor type of integral question.
Example – Find the partial fractions corresponding to the question x / (4–x)(x+5):
x / (4-x)(x+5)
=A / (4–x) + B / (x+5)
=A(x+5) + B(4–x) / (4–x)(x+5)
To find A & B, we will find the value of the coefficients of the numerators on both sides of the equation.
x = A(x + 5) + B(4 – x)
Now we can proceed by comparing the coefficients of similar powers of x on both sides. This is the best method for solving algebraic equations with ease following the Integration Rules.
Similarly when we substitute x = -5 in the equation, we get:
-5 = A(-5 + 5) + B(2 – (-5))
-5 = 0 + 7B
B = -5/7
Similarly, on substituting x = 4;
4 = A(4 + 3) + B(4 – 4)
4 = 7A + 0
A = 4/7
All we did to substitute the values of x is that we made the coefficients of A and B to be 0 separately. Our final result then will be:
x / (4–x)(x+5)=A(4–x)+B(x+5)
x / (4–x)(x+5)=4 / 7(4–x)– 5 / 7(x+5)
By this method similarly, the other integral problems of polynomial type can be easy to solve. Please note that in the above-given example, the degree of P was less than that of the degree of Q.
We hope you understood the right way to approach an integration problem, and you had a look at two of the most frequently used integration methods and the well defined Integration Rules.
Also Refer: What is Continuous Integration?