Integration of Sin2x: An introduction
In Mathematics, integration means the summation of given discrete data. Integration is done to find the function, given its derivative. The other use of integration is to calculate the area under the graph of a function. Integration is done to find the areas of the 2D region and the volumes of 3D objects. Integration is the opposite of differentiation. In the indefinite integration, we add the constant of integration. It is added to represent the constant term of the original function because we cannot obtain it through the anti-derivative process. The integral of sin2x and sin2x both is an indefinite integration because the limits are not defined here.
What is the Integration of sin2x?
The integral of sin2x is represented as \[\int {\sin 2xdx} \]. The given integral is calculated using the substitution method.
I = \[\int {\sin 2xdx} \]
Let u = 2x
Now, differentiating both sides, we get
du = 2dx \[ \Rightarrow dx = \dfrac{{du}}{2}\]
Substituting the value of u and du in the integral, we get,
\[\begin{array}{c}I = \int {\sin u{\rm{ }}\dfrac{{du}}{2}} \\ = \dfrac{1}{2}\int {\sin u{\rm{ }}} du\\ = \dfrac{1}{2}\left( { - \cos u} \right) + C\end{array}\]
Now convert the integral obtained in the terms of x,
\[I = \dfrac{{ - \cos 2x}}{2} + C\]
Thus, the integral of sin2x is \[\dfrac{{ - \cos 2x}}{2} + C\].
Integration of sin2x
The integral of sin2x is represented as \[\int {{{\sin }^2}xdx} \]. The given integral is calculated by first applying the formula of cos2x and then using the substitution method, the final answer is calculated.
I = \[\int {{{\sin }^2}xdx} \]
As we know, \[\cos 2x = 1 - 2{\sin ^2}x\].
Now find the value of sin2x from the above equation.
\[{\sin ^2}x = \dfrac{1}{2}\left( {1 - \cos 2x} \right)\]
Now, putting the value of sin2x in the integral, we get,
\[\begin{array}{c}I = \int {{{\sin }^2}xdx} \\ = \int {\dfrac{1}{2}\left( {1 - \cos 2x} \right)dx} \\ = \dfrac{1}{2}\int {\left( {1 - \cos 2x} \right)dx} \end{array}\]
The above integral can be broken into two integrals.
\[\begin{array}{c}I = \dfrac{1}{2}\int {\left( {1 - \cos 2x} \right)dx} \\ = \dfrac{1}{2}\int {dx} - \dfrac{1}{2}\int {\cos 2x{\rm{ }}dx} \\ = {I_1} - {I_2}\end{array}\]
Now integrating both the terms separately, we get
\[\begin{array}{c}{I_1} = \dfrac{1}{2}\int {dx} \\ = \dfrac{x}{2} + {C_1}\end{array}\]
\[{I_2} = \dfrac{1}{2}\int {\cos 2x{\rm{ }}dx} \]
Let u = 2x
Now, differentiating both sides, we get
du = 2dx \[ \Rightarrow dx = \dfrac{{du}}{2}\]
Substituting the value of u and du in the integral, we get
\[\begin{array}{c}{I_2} = \dfrac{1}{2}\int {\cos u{\rm{ }}\dfrac{{du}}{2}} \\ = \dfrac{1}{4}\int {\cos u{\rm{ du}}} \\ = \dfrac{1}{4}\left( {\sin u} \right) + {C_2}\end{array}\]
Now, convert the integral obtained in the terms of x.
\[{I_2} = \dfrac{{\sin 2x}}{4} + {C_2}\]
Now put the value of I1 and I2 in the original integral and find the value of I.
\[\begin{array}{c}I = {I_1} - {I_2}\\ = \dfrac{x}{2} + {C_1} - \dfrac{{\sin 2x}}{4} - {C_2}\\ = \dfrac{x}{2} - \dfrac{{\sin 2x}}{4} + C\end{array}\]
Thus, the integral of sin2x is \[\dfrac{x}{2} - \dfrac{{\sin 2x}}{4} + C\].
Solved Questions
1. How do we use substitution in integration?
Ans: We use a substitution method to change the independent variable from x to t. We assume that x = g(t). And substitute x = g(t) in the original function and then after differentiation, we substitute dx = \[g'(t)\].
2. Integration of an integrable function f(x) gives a family of curves differing by a constant value. True or false.
Ans: True, the integration of an integrable function gives F(x) + C which represents a family of curves as we change the value of constant C.
Practice Questions
1. How to find the definite integral of sin2x from 0 to \[\pi \]?
Ans: 0
2. How to find the definite integral of sin2x from 0 to \[\pi \]?
Ans: \[\dfrac{\pi }{2}\]
Interesting Facts
Integrals and derivatives are a part of a branch of Mathematics known as calculus.
Primitively while integration we used to divide the whole graph into rectangles of small width and added them together. This is known as the Riemann sum.
Integration is known as the anti-derivative.
Many rules are defined in substitution like the power rule, sum and difference rules, exponential rule, reciprocal rule, etc.
The fundamental theorem of calculus relates integration to differentiation.
Key Features
We add an arbitrary constant after integration to represent the constant term of the function that cannot be calculated by the antiderivative process.
The integral of sin2x is \[\dfrac{{ - \cos 2x}}{2} + C\].
The integral of \[sin^2x\] is \[\dfrac{x}{2} - \dfrac{{\sin 2x}}{4} + C\].
FAQs on Integration of sin 2x
1. What is the difference between definite integral and indefinite integral?
Definite Integral | Indefinite Integral |
1) A definite integral has an upper limit and lower limit. | 1) An indefinite integral has no upper limit and lower limit. |
2) It represents a particular solution of the given different solution. | 2) It represents a general solution or family of solutions of the given different solution. |
2. Write the name of 4 methods to solve an integration.
There are so many methods to solve an integration problem.
The most useful methods are listed below:
- Integration substitution method
- Integration by parts
- Integration using trigonometry identity.
- Integration of partial fractions
3. Where is integration used in real life?
In electrical engineering, integration is used to calculate the exact length of the cables to connect two substations that are miles apart from each other.
In Physics, integration is used to locate the position of the centre of mass, centre of gravity, etc.
In epidemiology, integration is used to study the spread of a contagious disease.
It is also used by space engineers to plan long missions.