Integration of Trigonometric functions: Explanation, illustration with examples, summary
In this article you will learn how to do the integration of trigonometric functions, let us first understand how is integral applied to the trigonometric functions.
How Integral is applied to the trigonometric function explained with examples?
Integral is a wide concept used in calculus for determining the area under the curve. Integral has wide use in mathematics, physics, and other engineering subjects.
Integral are of two types in calculus. Integration is a process of finding the integral of the function by using integral notation.
Integral calculus is very important in engineering. Integral is also calculated by using limits. Integral is also referred to as antiderivative. In integral the function is also in the form of trigonometry. Trigonometry is the measure of the triangle.
What is Integration?
Integration is a concept to calculate the area under the curve or the volumes of the curves or spheres. Integration is also stated as the inverting process of the derivative. Derivative and integration are the main topics of calculus.
Integration is further classified into two main types named definite and indefinite integrals.
When we have to calculate the functions without any limits, we use indefinite integral and when we have to calculate the limits of the function, we use definite integral in which upper and the lower limits are used.
Definite and indefinite integrals are not the same in working in general. Both the concepts have their working and both have the same calculations but in the last step, we apply limits in one of the concepts. Integral solve all kinds of problems by using these two concepts.
In these types of integrals, the trigonometric function is applied in a wide range. There are many general formulas for trigonometry for use in integration. We can calculate the integration of trigonometric functions.
How Integration of trigonometric functions is done?
Trigonometry functions are widely used in calculus for the measurements of different geometrical objects. Trigonometry functions are also applied in derivatives, the results in the integral of the trigonometric functions are inverse as compared to the differential.
Integral is applied to a single trigonometric function as well as the group of the trigonometric functions. In integral, trigonometry functions must be calculated according to the required desires.
Example 1
Find the integral of cos√x + sin√x – tan√y with respect to x
Solution
Step 1: Write the given function along with the integral notation.
ʃ (cos√x + sin√x – tan√y) dx
Step 2: Apply the laws of the integration.
ʃ (cos√x + sin√x – tan√y) dx = ʃ (cos√x) dx + ʃ (sin√x) dx – ʃ (tan√y) dx
Step 3: Take the constant outside the integral notation.
ʃ (cos√x + sin√x – tan√y) dx = ʃ (cos√x) dx + ʃ (sin√x) dx – (tan√y) ʃ dx
ʃ (cos√x + sin√x – tan√y) dx = I 1 + I 2 – I 3
Step 4: Now solve the integrals one by one.
I 1 = ʃ (cos√x) dx … (1)
let √x = w, apply differential on both sides.
1/2w = dw/dx
dx = 2wdw
put these terms in (1).
I 1 = ʃ (cos√x) dx = ʃ cos(w) 2w dw
= 2 ʃ w cos(w) dw
Apply product law.
ʃ u * v = uʃv – ʃ d/dx(u) ʃv
= 2 ʃ w cos(w) dw = 2(w sin(w) – ʃ (1) (sin(w)) dw
= 2wsin(w) – ʃ sin(w) dw
= 2wsin(w) – (-cos(w))
= 2wsin(w) + cos(w)
Put the value of w.
I 1 = ʃ (cos√x) dx = ʃ cos(w) 2w dw = 2√x sin(√x) + cos(√x) + c 1
Similarly, for I 2
I 2 = ʃ (sin√x) dx = ʃ sin(w) 2w dw = -2√x cos(√x) + sin(√x) + c 2
Step 5: Put the values of I1, I2, and I3.
ʃ (cos√x + sin√x – tan√y) dx = I1 + I2 – I3
ʃ (cos√x + sin√x – tan√y) dx = 2√x sin(√x) + cos(√x) + C1 -2√x cos(√x) + sin(√x) + C2 – xtan√y + C3
Step 6: In the above equation c1, c2, and c3 are arbitrary constants, take C = min (C1, C2, C3)
ʃ (cos√x + sin√x – tan√y) dx = 2√x sin(√x) + cos(√x) – 2√x cos(√x) + sin(√x) – xtan√y + C
These types of integral questions need lengthy calculations to reach the final equation. To avoid such large calculations, you can use integrate calculator for the process of integration.
Example-2
Find the integral of sin3(x) * cos(x) with respect to x
Solution
Step 1: Write the given function along with the integral notation.
ʃ (sin3(x) * cos(x)) dx
Step 2: Use insertion method and put sin(x) = w.
When sin(x) = w
Take derivative on both sides.
d/dx (sin(x)) = d/dx (w)
cos(x) = dw/dx
cos(x) dx = dw
Step 3: Put these terms in the equation.
ʃ (sin3(x) * cos(x)) dx = ʃ w3 dw
Step 4: Apply the power law.
ʃ (sin3(x) * cos(x)) dx = w3+1/ 3 + 1 + C
ʃ (sin3(x) * cos(x)) dx = w4/4 + C
Step 5: Now put the value of w = sin(x), in the above equation.
ʃ (sin3(x) * cos(x)) dx = sin4(x)/4 + C
Example-3
Find the integral of sin(x) * cos3(x) with respect to x
Solution
Step 1: Write the given function along with the integral notation.
ʃ (sin(x) * cos3(x)) dx
Step 2: Use insertion method and put sin(x) = w.
When cos(x) = w
Take derivative on both sides.
d/dx (cos(x)) = d/dx (w)
-sin(x) = dw/dx
-sin(x) dx = dw
Step 3: Put these terms in the equation.
ʃ (sin(x) * cos3(x)) dx = – ʃ w3 dw
Step 4: Apply the power law.
ʃ (sin(x) * cos3(x)) dx = -w3+1/ 3 + 1 + C
ʃ (sin(x) * cos3(x)) dx = -w4/4 + C
Step 5: Now put the value of w = sin(x), in the above equation.
ʃ (sin(x) * cos3(x)) dx = -cos4(x)/4 + C
Summary
In this article you learned what is integral calculus and how to do the integration of trigonometric functions, we further discussed some examples which illustrate the method of integration of trigonometric functions. If you have any doubt please ask us in the comment sections. If you like our content please let us know in the comment section. We appreciate your honest reviews.