Integration of Trigonometric Functions

Integration is nothing but a mathematical approach, at which point you involve in using the functions and put all the functionalities back together. The process of finding the area under the curve is known as integration.

What is Integration?

Integration is the concept of determining the area or volume of the curve. We would have studied derivation; the inverse approach to derivation is integration. This is one of the central concepts involved in calculus.

Integration can be classified as two:

(i) Definite Integrals

(ii) Indefinite Integrals

• The integrals with both the start and end values are called definite integrals.

• The integrals with no start and no end value are called indefinite integrals.

The function has limits and the concept of determining the integral using the mentioned limits, and then it is called a definite integral. In comparison, the indefinite integral has no limits. We have to determine the limits of the integral on our own using various formulas.

To calculate the integral, we use trigonometric functions. Whether it can be definite or indefinite, the trigonometric formula helps resolve all the problems. We’ll see the usage and how the trigonometric function works in integrals in the below sections.

What is a Trigonometric Function?

Integral has outspread its usage, especially in maths and physics engineering subjects. Integration is nothing but determining the integral function using various integral notations mentioned as the antiderivative. The two types of integrals, which we have seen in the above section, are used in integration to find the integral functions. 

Integration of Trigonometric Functions: Importance

Trigonometry is used to measure the objects of different geometries. Derivatives are also used in trigonometric functions since they are a significant calculus part. And we know that integration is the inverse process of derivations.

Integral has also been practised in single as well as group trigonometric functions. We can see a few examples to get accurate knowledge of the integration of trigonometric functions.

This segment will consider the integration used in trigonometric functions. Integration is nothing but the actual process of determining the integral. 

• There are a few formulas that you can use for fast calculations. We can also implement the formula in trigonometry to get an accurate solution. 
• The formula has various functionalities starting from the sine and cosine. 
• To understand all these categories and techniques, we should derive some valid examples.

The integration formula is used even in resolving algebraic problems; the study says that in most cases, integration formulas are widely used in logarithmic functions and trigonometric ratios. 

Below are the sample examples with solutions; you can solve the problems in simple steps using the trigonometric integration formula.

Example  of integration of trigonometric functions questions

Example A

Integrate  ʃ sin 5x dx

Solution : Let’s consider 5 x = a

a = 5x  ——(1)

Applying the differentiation in equation (1)

da = 5dx,

we can rewrite this as :  (1/5)da = dx

Substituting the same in the problem,

ʃ sin 5x dx = ʃ sin a (1/5)da

                  = (1/5) ʃ sin a da

Applying the antiderivative rule,

= (1/5) (-cos u) +C

=-1/5 cos 5x+C

Example B

Calculate: ʃsin2 x cos 8  x dx   

Solution: ʃsec5 x tan 8 x dx   = ʃ(sec2 x)2 tan 8  x  sin x dx   = ʃ(1-tan2 x)2 tan 8  x  sec x dx   

consider u = tan x, we can come to the derivation as, du= -sec x dx

ʃ(sec5 x) tan 4  x  sec x dx   = ʃ(1-u2 )2 u 8  x  du

                                             = ʃ(-u8  +2u10  -u12  ) du

                                       = -u9 /9+2u11 /11 – u13/13 + C

                                       =-tan9 x/9 +2 tan11  x /11- tan13 /13  + C

Example C:

ʃ ex  sin (ex ) dx   

Solution :  Let’s consider u = ex

Applying the differentiation,

du= ex dx,

we can rewrite this by Substituting the same in the problem,

ʃ ex  sin (ex ) dx   = ʃ sin (ex ) ex dx   

= ʃ sin u du  

= cos u + C

=  cos (ex ) + C

Example D:

Estimate the following integral ∫x2 secx3dx.

Solution: ∫x2 secx3dx = ∫ secx3 x2 dx

Set u = x3 and du = 3x2dx or du/3 = x2dx, then we have:

∫x2 secx3 dx

= ∫sec u du/3

= 1/3 * ∫sec u du

= 1/3 *(-tan u) + C

= 1/3 *(-tan x3) + C

Example E:

Evaluate ∫(8 cos x 5tan2x) dx

Solution:  ∫(8cos x 5tan2x) dx

= 8∫cosxdx – 5∫tan2x dx

= -8sinx – 5secx + C

Example F:

Determine the following integral ∫x2 sec x3dx.

Solution:

∫x2 sec x3dx = ∫ sec x3 x2 dx

Set u = x3 and du = 3x2dx or du/3 = x2dx, then we have:

∫x2 sec x3 dx

= ∫sec u du/3

= 1/3 * ∫sec u du

= 1/3 *(-tan u) + C

= 1/3 *(-tan x3) + C

The above examples explain the trigonometric functions in a significantly straightforward way. In simpler words, integration combines the part of every functionality and puts it together to form an equation and derives the solution by following a finite process. Trigonometric functions are applied in various sectors to derive an accurate solution for measuring things. Using integrations helps resolve tedious problems and measures the diverse number of items present in the real world.

Conclusion

Integration is the reversal method of differentiation. In the calculation of differentiation, we use a function that is provided and determine the derivation of the function. Whereas in integration, the differential will be provided, we should derive the function. And that’s why we call integration the inverse method of differentiation. The above examples would have given a clear idea about how the integration of trigonometric functions works.