Integrals: Types, Formulas, and Properties

Integrals are the anti-derivatives of a function determined through the process of Integration. Integrals are the fundamental object of Calculus and are the representation of the area of a region under a curve. 

• Integrals are an integral concept of Calculus along with Derivatives. 
• They are also known as antiderivative and primitive of a function.
• Integration is the process of computing an integral or anti-derivative. 
• Integrals are of two types namely Definite Integrals and Indefinite Integrals.

Integrals assign numbers to functions that determine area and volume problems, displacement and motion problems, etc. that arise by combining all the small data.

Read More: NCERT Solutions For Class 12 Mathematics Integrals

Key Terms: Integrals, Integral Calculus, Integration, Definite Integral, Indefinite Integral, Integration by Parts, Integral Formulas, simpsons rule

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What are Integrals?

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Integral is the representation of the area under the curve. Integrals are also referred to as anti-derivatives.

• F(x) is denoted as an antiderivative or Newton-Leibnitz Integral.
• It is the primitive of a function f(x) on an interval I.
• For every value of x in I, F'(x) = f(x).
• The actual value of an integral can be obtained (approximately) by drawing rectangles.
• Definite Integral of a function is the area of the region bounded by the graph of the given function between two points in the line.
• The integral of a function is specified over an interval on which the integral is defined.

Integrals Detailed Video Explanation

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Types of Integrals

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Integral Calculus helps to solve two major types of problems:

1. The problem of finding a function if its derivative is given.
2. The problem of finding the area bounded by the graph of a function under given conditions.

Thus, Integrals can be divided into two major categories: 

1. Indefinite Integrals: Indefinite integral is a function that takes the ant-derivative of another function.
2. Definite Integrals: Definite integral is defined to be exactly the limit and summation to find the net area between a function and the x-axis

Indefinite Integral

Indefinite integrals do not have a pre-existing value of limits, i.e. they are not defined using the upper and lower limits. Indefinite integrals represent the family of a function whose derivatives are f, and thus, it returns a function of the independent variable.

The representation of the integration of a function f(x) given by F(x) is as follows:

Where

• R.H.S. of the Equation denotes Integral of (x) with respect to x.
• F(x) is the Anti-derivative or Primitive.
• f(x) is the Integrand.
• dx is the Integrating Agent.
• x is the Variable of Integration.
• C is the Constant of Integration.

Indefinite and Definite Integrals

Definite Integral

Definite integrals have a pre-existing value of limits. It is a type of integral that has upper and lower limits. It is also referred to as a Riemann Integral when it is bound to lie on the real line.

Definite Integrals are represented as:

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Fundamental Theorem of Integral Calculus

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Integrals are the function of the region covered by the curve y = f(x), a ≤ x ≤ b, x-axis, and the ordinates x = a and x = b, where b>a.

Let us assume that x is a given point in [a,b]. Then the area of a function is represented by \(\int\limits_a^b f(x) dx\).

There are two fundamental theorems of integral calculus led by the concept of area function:

• First Fundamental Theorem of Integral Calculus
• Second Fundamental Theorem of Integral Calculus

First Fundamental Theorem of Integrals

According to the first fundamental theorem of integrals, when the function is continuous on [a,b], then

  Thus, A'(x) = f(x) for all x ϵ [a,b].

Second Fundamental Theorem of Integrals

When f is a continuous function of x stated on the closed interval [a,b] and F is another function such that d/dx F(x) = f(x) for all x in the domain of f, then, 

 It is referred to as the definite integral of f over the range [a,b], where a is the lower limit and b is the upper limit.

Read More: Integration Rules

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Integrals Formulas

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The list of integral formulas similar to differentiation formulas are listed below:

• ∫ xⁿ dx=xⁿ⁺¹ /n+1+C, n ≠ -1
• ∫ dx = x + C
• ∫ cosx dx = sinx + C
• ∫ sinx dx = -cosx + C
• ∫ sec²x dx = tanx + C
• ∫ cosec²x dx = -cotx + C
• ∫ sec²x dx = tanx+C
• ∫ secx tanxdx = secx + C
• ∫ cscx cotx dx = -cscx + C
• ∫1/(√(1-x²)) = sin^-1 x + C
• ∫-1/(√(1-x²)) = cos^-1 x + C
• ∫1/(1+x²)= tan^-1 x + C
• ∫-1/(1+x²)= cot^-1 x + C
• ∫1/(x√(x² -1)) = sec^-1 x + C
• ∫-1/(x√(x² -1)) = cosec^-1 x + C
• ∫ e^xdx = e^x + C
• ∫dx/x = ln|x| + C
• ∫ a^x dx = a^x/ln a + C

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Methods to Find Integrals

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Integrals of a function can be found using three prominent methods of integration which are as follows:

• Substitution Method
• Integration by Parts
• Integration by Partial Fractions

Substitution Method

Some integrals can be found easily by the substitution method. If u is given as a function of x, then u' = du/dx.

Here, u = g(x).

Integration by Parts

Integration by Parts is a special method of integration that is often useful when two functions are multiplied together.

Integration by Partial Fractions

Integration of rational functions whose numerator and denominator have positive integral powers of x with constant coefficients can be done by resolving them into partial fractions.

In order to find ∫ f(x)/g(x) dx, the improper rational function will be decomposed into a proper rational function and then integrated.

Here g(x) = a(x) . s(x)

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Properties of Integrals

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The properties of integrals are as follows:

• Integrand itself is the derivative of an integral.

∫ f(x) dx = f(x) +C

• Two indefinite integrals that have the same derivative are equivalent and lead to the same family of curves. 

∫ [ f(x) dx – g(x) dx] = 0

• The integral of the sum or difference of a finite number of functions is equivalent to the sum or difference of the integrals of the individual functions. 

∫ [ f(x) dx + g(x) dx] = ∫ f(x) dx + ∫ g(x) dx

• The constant is placed outside the integral sign.

∫ k f(x) dx = k ∫ f(x) dx; k ∈ R

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Uses of Integrals

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Integrals are used for mainly two applications:

1. First, integrals are used to calculate f from f’.
2. If a function f is differentiable in the interval being considered, then f’ is defined.
3. In differential calculus, the derivatives of a function can be undone with the help of integral calculus.
4. Secondly, integrals are used to calculate the area under a curve.

Applications of Integral Calculus

Integral Calculus is used in the following problems:

• Work
• Area between Two Curves
• Centre of Mass
• Surface Area
• Distance, Acceleration, and Velocity
• Average Value of a Function
• Volume
• Kinetic Energy

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Integrals Solved Examples

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Given below are a few solved examples on Integrals: 

Example 1: Determine the integral of cos²n with respect to n.

Solution: Assume that f(n) = cos²n 

As we know, 2 cos²A = cos 2A + 1

Thus, f(n) = (1/2)(cos 2n + 1)

Now, we will find the integral of f(n).

∫f(n) dn = ∫(1/2)(cos 2n + 1) dn = (1/2) ∫(cos 2n + 1) dn

= (1/2) ∫cos 2n dn + (1/2)∫1 dn = (1/2) (sin 2n/2) + (1/2) n + C

= (sin 2n/4) + (n/4) + C = (1/4)[sin 2n + n] + C

Thus, the integral of cos²n with respect to n is (1/4)[sin 2n + n] + C.

Example 2: What will be the integral of f(x) = √x?

Solution: It is given that, 

f(x) = √x

∫f(x) dx = ∫√x dx

\(\int \sqrt{x}\ dx = \int x^{\frac{1}{2}}\ dx\)

As we know that, 

\(\int x^{n}\ dx = \frac{x^{n+1}}{n+1}+C\)

Proceeding further, we get

\(\int \sqrt{x}\ dx = \frac{x^{\frac{1}{2}+1}}{\frac{1}{2}+1}+C\)

\(= \frac{x^{\frac{1+2}{2}}}{\frac{1+2}{2}}+C\)

\(=\frac{2}{3}x^{\frac{3}{2}}+C\)

Thus, the integral of  f(x) = √x is \(\frac{2}{3}x^{\frac{3}{2}}+C\).

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Things to Remember

• Integrals are the primitive value of function found by the process of integration.
• It is a mathematical object that is interpreted as an area or a generalization of an area.
• Integrals and Derivatives are the two fundamental objects of Calculus.
• Definite Integrals and Indefinite integrals are the two types of integrals in calculus.
• When a polynomial function is integrated, the degree of integral increases by 1.
• Integrals are found through substitution, integration by parts, and integration by partial fractions.
• Integrals are used in various fields of engineering and three-dimensional models.

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Previous Years’ Questions

1. \(\int \frac{\sin 2 x}{\sin ^{4} x+\cos ^{4} x} \)is equal to... (BITSAT - 2011)
2. Let [.] denote the greatest integer function then the value of… (AIEEE - 2011)
3. \(\displaystyle\int_{1/2}^{2}|\log_{10}\,x|dx=\)...
4. Evaluate: \(\int\frac{sin x}{sin 4x} dx\)... 
5. Evaluate: \(\int\frac{x^{3}+x}{x^{4} 9}dx\)… 
6. If a is a positive integer, then the number of values... 
7. The integral ∫cos (logx) dx is equal to… (JEE Main - 2019)
8. The number of integral terms in the expansion…
9. \(\int\limits_{0} ^{1}\frac{dx}{[ax+b(1 x)]^2}\) is equal to…
10. Prove that the value of the integral... (JEE Advanced - 1988)