Integration of Trigonometric Functions

Just like in geometry we study about shapes, in algebra, we study about arithmetic operations. Similarly, calculus is about continuous change. Calculus is divided into two parts called: (i) differential calculus (ii) integral calculus

Differential calculus is also known by the names, differentiation and derivative.

Integral calculus is also known by the names, integration and antiderivative. So what is Differentiation? Differentiation is the study of the rate of change in quantities. Geometrically speaking, when we find the derivative of a point on a curve it is the slope of the tangent line. To find the derivative of a function it’s necessary that a derivative exists and is defined at the point

For example:

F(x)= x²+x-2=0

F’(x)= 2x+1.

Application of Differentiation:

Differentiation is used in the following:

a) For optimization

b) Physics

c) Differential Equations

d) Mean Value Theorem

The reverse process of differentiation is integration.

Integration:

Integration is a mathematical concept wherein you join and put things back together. Geometrically speaking, integration is a process to find the area under the curve. It is represented in the following way:

∫f(x) dx

where ∫ is a symbol for integration, f(x) is the integrand and ‘x’ is the integration variable.

Integration can be done of two type integrals namely,

(i) Definite Integrals

(ii) Indefinite Integrals

Definite integrals are those integrals which have a start and an end value. This means that the curve that’s being discussed is within an interval [a, b].

Indefinite Integrals are those integrals which do not have a start and an end value.

For example:

∫2x + 1 dx

=x²+x+c, where c is a constant function.

There are various ways of integrating a function. These ways are:

a) Integration by Substitution

b) Integration by Parts

c) Integration of Trigonometric Functions

d) Integration of Some Particular Function

e) Integration by Partial Fraction

We will talk about every method of integration in detail.

Integration by Substitution

There are times when the given function is a little complicated and thus, making it difficult for us to integrate. To make it easy we use a different independent variable to make it easier to integrate. This is known integration by substitution.

∫sin(1-x)(2-cos(1-x)) dx

u = 2 - cos(1 - x)  du = -sin(1 - x)dx   ⇒sin(1 - x)dx = -du

∫sin(1-x)(2-cos(1-x)) dx = -∫u du

= - ⅕ u + c

Integration by Parts

Integration by parts requires the integrand function to be the multiple of two more functions.

Let’s say we have an integrand function to be f(x).g(x).

Integration by parts is represented as:

∫f(x).g(x).dx = f(x).∫g(x).dx–∫(f′(x).∫g(x).dx).dx

A rule is followed while integrating functions by the method of integration by parts. It’s called ILATE. It stands for inverse trigonometry, logarithm, algebra, trigonometry and exponents.

Integration of Trigonometric Functions

While integrating a function, if trigonometric functions are present in the integrand we can use trigonometric identities to simplify the function to make it simpler for integration. Some integration formulae of trigonometric functions are given below:

Sin2x= $\frac{1-cos2x}{2}$

cos2x= $\frac{1+cos2x}{2}$

4sin3x=3sinx–sin3x

4cos3x=3cosx+cos3x

Mentioning integration of all trigonometric functions will be very difficult since there are many formulae.

Integration of Some Particular Function

Some functions when being integrated involve the usage of essential formulae of integration. It is applied to our given function to have it in a standard form of the integrand.

Integration by Partial Fraction

Integrals can also be represented in p/q form, the way rational numbers are expressed. By expressing integrals in this way we can use some special formulae to easily integrate the given function.

Definite Integrals

The major purpose of integration is to find the area under the curve. You can find the area of definite integrals only. Other than using integration, there’s another to find the area which is called Limit of Sums.

The formula for limit of sum is:

$\int_{a}^{b}$ f(x) dx = (b - a) $\lim_{n\rightarrow∞ }$ (1/n)[f(a) + f(a + h) + … + f(a + {n - 1}h)]

Definite Integrals have several properties which are used while integrating. They are discussed below:

1. $\int_{a}^{b}$ f(x) dx = $\int_{a}^{b}$ f(t) d(t)

2. $\int_{a}^{b}$ f(x) dx = -$\int_{a}^{b}$ f(x) dx

3. $\int_{a}^{a}$ f(x) dx = 0

4. $\int_{a}^{b}$ f(x) dx = $\int_{a}^{c}$ f(x) dx + $\int_{c}^{b}$ f(x) dx

5. $\int_{0}^{a}$ f(x) dx = $\int_{0}^{a}$ f(a - x) dx

Solved Example

Evaluate the given below integral?

Solution

Using integration by substitution

∫ sin(1 - x)(2 - cos(1 - x))

u = 2 - cos(1 - x)  du = -sin(1 - x)dx     ⇒sin(1 - x)dx = -du

∫sin(1 - x)(2 - cos(1 - x))

= - ⅕ (2 - cos(1 - x))

= -⅕(2 - cos(1 - x))

Fun Facts:

• You can easily check the answer you got after integrating a function.

• Differentiate the integrated function, if you get your question again it means you are right.

• The symbol of integration ‘∫ ’ is a large “S” which stands for summa.

• Summa is Latin for a sum. This was first used by Gottfried Wilhelm Leibniz.

Q1. Can a Function be Integrated in More than One Way? Mention Some Trigonometric Integration Formulae.

Ans. Yes, a function can be integrated in more than one way. It depends on us which method is comfortable for us. As discussed above there are several ways of integration. Like:

Integration by Substitution

Integration by Parts

Integration Using Trigonometric Identities

Integration of Some Particular Function

Integration by Partial Fraction

Like any math problem, they are several ways of integrating a function. No matter what method you choose you will get the same answer. Like everything, there are always some exceptions. Similarly, in integration some functions are integrated only by following some selected methods that is, there are only a few specific ways it can be solved. For example:

The integral of Log x can only be found in a limited number of ways.

Some examples of trigonometric integration formulae are:

∫sinx = -cosx + c

∫Cosx = sinx + c

∫sec2x = tanx + c

Q2. What is ‘c’ in Integration? Is it Important?

Ans. ‘c’ refers to a constant function. A constant function is a function in which no matter what value you input you will always get the same output. We already know that integration is a reverse process of differentiation. So when a constant function is differentiated it becomes zero. Thus, when taking the same function for integrating we always add ‘c’ for the unknown constant function. We put ‘c’ only in indefinite functions. Definite functions do not have ‘c’ because they have a defined start and end. Therefore, they do not require a constant function. So, for indefinite integrals, it is extremely important to add ‘c’ in the end.