What is Integral Calculus: List, Types and Questions

Integral calculus is used for two broad purposes. The first purpose is to find the function when the derivative of the function has been provided. The second is to find the area bounded by the curve of a function between two points, or the general formula of the area bounded by a function. The topic of integral calculus contains many formulae that must be understood by the student. From the examination point of view, integral calculus can be up to 20 marks, forming 1, 2, 4, and 6 mark questions. It generally forms a significant part of the CBSE board Class XII syllabus .

1. The Definite Integral

There are two types of integrals. The first among these is a definite integral. The definite integral is more widely used in real life and has more tangible physical implications. A definite integral is an integral which is bounded by two points. It has an upper limit and a lower limit, and the integral is calculated between these limits. On a Cartesian plane, this can be expressed as the area bounded by a curve between two points on an axis. The result of a definite integral is generally not a formula, but a constant number, unless one of the limits is a variable.

A definite integral is like water below a curve y=f(x) from x=a to 𝑥=b. You are trying to measure how much you have filled. This type of integral will measure the Area above the axis (where 𝑓(𝑥)≥0), which counts positively. Area below the axis (where 𝑓(𝑥)<0) counts as negative. The result will be zero if the positive and negative parts cancel. It connects anti-differentiation to the following areas in two steps:

1. Antiderivative link
   If $F\text{'}\left(x\right)=f\left(x\right)$, then ${\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$.
2. Existence
   Every continuous f on [a,b] has a definite integral.

Properties of Definite Integrals

2. The Indefinite Integral (The Antiderivative)

The second type of integral is known as an indefinite integral. An indefinite integral does not have an upper limit or a lower limit. It cannot be expressed completely on a Cartesian plane and provides the general function of the area bounded by a curve. The result of an indefinite integral is generally a function in terms of an independent variable, followed by an arbitrary constant. A definite integral is basically the difference between the values of an indefinite integral at the upper and lower limits. Do remember that IISER and CUET exams ask implementation-based questions on this type of integral.

This integral's purpose is to reverse the process of differentiation. It doesn't calculate a value; it finds a function. Indefinite integral aims to answer "What function, F(x), when I take its derivative, gives me the function f(x)?". This will result in a family of functions.

Given a function f(x), an indefinite integral finds all functions F(x) whose derivative is f(x):

$$F\text{'}\left(x\right)=f\left(x\right)$$

Therefore,

$$\int f\left(x\right)dx=F\left(x\right)+C$$

Since the derivative of any constant (like 5, -10, etc.) is zero, there are infinite possible answers. We represent this ambiguity with + C. If f(x) is the recipe for the slope of a hill at any point, the indefinite integral F(x) + C is the recipe for the shape of all possible hills that could have those slopes.

The following are the integrals that are useful for JEE Main and IIT JAM exam aspirants:

In differential calculus, we have to find the derivative or the differential of a given function. However, in integral calculus, one has to find a function whose differential is provided. Integration is the inverse of differentiation. C is the integration constant. Integration is the process that refers to the inverse of differentiation. Considering ∫f(x) dx = F(x) + C, these types of integrals are known as General Integrals or Indefinite Integrals. Where C is the arbitrary constant that differs with multiple antiderivatives for the given function. 

Note: a derivative of the function is specified, but a function can have many integrals or antiderivatives.

First, let us begin by understanding the symbols and the terms we use in integration.

Integral calculus is the branch of analysis that covers theory, technique, and applications to systematically study integrals and related properties. Definite integrals and indefinite integrals form a part of the integral calculus. Integration, on the other hand, is a mathematical operation that computes an integral of a given function f. It has two forms including definite and indefinite integration. In simple words, integration is just a computation step to apply techniques for obtaining a numerical value/ family of antiderivati

These are important types of questions on integral calculus that students of the NEET entrance examination must practice

1. Calculate the integral ₀∫⁵⁰(x²+4x+2)dx

Solution.

₀∫⁵⁰(x²+4x+2)dx
=₀∫⁵⁰(x²)dx+₀∫⁵⁰4xdx+₀∫⁵⁰2dx

=(50³-0³)/3 + 2(50²-0²)+2(50-0)

= 125000/3 + 5000 + 100

= 46,766.66

2. Calculate the integral ∫ (x² +sinx)dx

Solution.

∫ (x² +sinx)dx

= ∫ (x²)dx+(sinx)dx

= x³/3 - cosx + c

3. Calculate the integral $$\int \frac{x}{\sqrt{1-x^2}}dx$$

Solution. 

∫x/√1-x²dx

Let x=sinx

dx=cosxdx

∫x/√1-x²dx = ∫(sinx/cosx)cosxdx

=∫ sinx dx

=cosx+c

=√1-x²+c