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Integration of Trigonometric Functions – Formulas, Solved Examples

Integration of Trigonometric functions involves basic simplification techniques. These techniques use different trigonometric identities which can be written in an alternative form that are more amenable to integration.

Integration of Trigonometric Functions - Formulas, Solved Examples
Integration of Trigonometric Functions – Formulas, Solved Examples


Representation

The integration of a function f(x) is given by F(x) and it is represented by:

∫f(x)dx = F(x) + C

Here,

R.H.S. of the equation means integral f(x) with respect to x.

F(x) is called anti-derivative or primitive.

f(x) is called the integrand.

dx is called the integrating agent.

C is called constant of integration or arbitrary constant.

x is the variable of integration.

Also, check integral formulas here.

Integration of Trigonometric Functions Formulas

Below are the list of few formulas for the integration of trigonometric functions:

  • ∫sin x dx = -cos x + C
  • ∫cos x dx = sin x + C
  • ∫tan x dx = ln|sec x| + C
  • ∫sec x dx = ln|tan x + sec x| + C
  • ∫cosec x dx = ln|cosec x – cot x| + C = ln|tan(x/2)| + C
  • ∫cot x dx = ln|sin x| + C
  • ∫sec2x dx = tan x + C
  • ∫cosec2x dx = -cot x + C
  • ∫sec x tan x dx = sec x + C
  • ∫cosec x cot x dx = -cosec x + C
  • ∫sin kx dx = -(cos kx/k) + C
  • ∫cos kx dx = (sin kx/k) + C
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To understand this concept let us solve some examples.

Integration of Trigonometric Functions Examples

Example 1:

Question- Integrate 2cos2x with respect to x.

Solution- To integrate the given trigonometric functions we will use the trigonometric identity

cos2x=(1+cos2x2)

Form this identity

2cos2x=1+cos2x

Substituting the above value in the given integrand, we have

2cos2xdx=(1+cos2x).dx(1)

According to the properties of integration, the integral of sum of two functions is equal to the sum of integrals of the given functions, i.e.,

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

Therefore equation 1 can be rewritten as:

(1+cos2x)dx=1dx+cos2xdx

=x+sin2x2+C

This gives us the required integration of the given function.

Example 2:

Question- Integrate sin 4x cos 3x with respect to x.

Solution- To integrate the trigonometric function we will use the trigonometric identity:

sinxcosy=12[sin(x+y)+sin(xy)]

Form this identitysin4xcos3x=12(sin7x+sinx)

Therefore,

(sin4xcos3x)dx=12(sin7x+sinx)dx

From the above equation we have:

12(sin7x+sinx)dx=12(sin7x+sinx)dx(ii)

According to the properties of integration, the integral of sum of two functions is equal to the sum of integrals of the given functions, i.e.,

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

Therefore equation 2 can be rewritten as:

12(sin7x)+12(sinx)dx

=cos7x14+cosx2+C

This gives us the required integration of the given function.

Example 3:

Question- Integrate sin2 x. cos2x.

Solution- Before integration let us use few trigonometric relations in order to simplify the integrand.

We know,  2sinxcosx=sin2x

sinx.cosx=sin2x2

Substituting the value in the given integrand, we have

sin2x.cos2xdx=(sinx.cosx)2dx=(sin2x2)2

=14sin22x(i

Also we know, sin2x=1cos2x2

Substituting the above value in equation (i), we have

14sin22x=141cos4x2

=18dxcos4x8dx

=18x+C1sin4x32+C2

=18xsin4x32+C

Video Lesson on Trigonometry

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Integration of Trigonometric Functions Questions

Try solving the following practical problems on integration of trigonometric functions.

  1. Find the integral of (cos x + sin x).
  2. Evaluate: ∫(1 – cos x)/sin2x dx
  3. Find the integral of sin2x, i.e. ∫sin2x dx.

To learn more about trigonometry and Integration of function, download BYJU’S-The Learning App and experience the fun in learning.

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