Calculus is one of the ancient and most widely used branches of mathematics, this branch is widely used and taught to students who have a high interest in mathematics and want to pursue their career in this field. It is one of the vital topics to be known for engineering. The concept of this topic although seems difficult in the beginning as it requires a unique concept to solve the problems, but can be easily mastered upon with regular hard work and grit. There is a wide range of topics that one studies in calculus like integration formula, differentiation, and differential equations.
This article discusses integration and all the integration formulas and methods which one has to remember to solve the basic as well as the advanced questions of calculus. First of all, let’s discuss integration. In simple words, Integrals can be defined as the antiderivative of a given function. One can define integration as the area of the given region under a given curve. There are two types of integrals that we study in mathematics.
- Definite integral: It is a type of integration that helps in finding out the area under the particular region, that is when the limits of the integration are already provided. It is the area of the region that is already given between the graphs. After solving the integration, one needs to put the limit and take out the results by subtracting the upper limit from the lower limit.
- Indefinite integration: On the other hand Indefinite integration is the type of integration in which one needs to solve the problem simply, by just applying simple rules and formulas which one has already studied in the beginning, according to Lamar.edu.
Integration and its formulas are something that one should always remember to solve questions. Here are some of the main methods which one study in integration, these are:
- Finding the integrals with the substitution method: In this method, one needs to select the complex term and find the best possible substitute for the same to make the question simple and solvable. Let’s say u’ = du/dx, so
∫ f(u)u’ dx = ∫ f(u)du, here u = g(x).
- The next way to find the integral is to solve them with by-parts. This method of finding integral is applied when the functions are of the product form. One needs to use this method taking function at a time or this can be done with the help of the formula given below:
∫f(x)g(x) dx = f(x)∫ g(x) dx – ∫ (f'(x) ∫g(x) dx) dx.
- The last method to find out the integral is by partial integration. This method is performed when the integration of rational functions that consists of numerator and denominator as the position power of x. This is done by resolving them into the partial fraction.
With the help of the formula given below, how one can find out the integral using partial integration. As one can understand in a better way with the help of this formula
∫f(x)/g(x) dx = ∫ p(x)/q(x) + ∫ r(x)/s(x), here g(x) = a(x) . s(x)
These are some of the methods to find out the Integrals. The application of integrals is many, one can use integration to find out the area of the region like a sphere or a square and the volume of cube or cylindrical bodies too, Integration is also used to find out the distance with the help of velocity.
Integral calculus is a vital topic that should be cleared from the beginning. If students face any doubts while solving the problems then they can visit the website of Cuemath to solve the doubts as well as learn all his weak concepts.