Question

How do you find the integral of $\sin \left( 2\pi t \right)dt$ ?

Answer 1

In order to solve the above question, we have to apply trigonometric substitutions, First we have to make a few substitutions so that the given integral is simplified After that we will integrate term by term using simple integration formulas.

Complete step by step answer:
The above question belongs to the concept of integration by trigonometric substitution. Here we have to use basic trigonometric substitutions in order to integrate the given function. We have to integrate $\sin \left( 2\pi t \right)dt$.
We will first make a few substitutions.
Our first step is to let $2\pi t=u$
$\Rightarrow du=2\pi dt$
In order to convert the derivative in our integral expression we have to multiply the function inside the integral with $2\pi$ therefore, we have to balance the additional change. So, we will divide the whole equation by $2\pi$
Now replacing t in the given integral and transforming the integral in terms of the substitution.

& \int{\sin \left( 2\pi t \right)dt}=\dfrac{1}{2\pi }\int{\sin \left( 2\pi t \right).2\pi dt} \\\ & \Rightarrow \int{\sin \left( 2\pi t \right)dt}=\dfrac{1}{2\pi }\int{\sin \left( u \right)du} \\\ & \Rightarrow \int{\sin \left( 2\pi t \right)dt}=-\dfrac{1}{2\pi }\cos (u)+C \\\ \end{aligned}$$ After applying the integration rule, we get. $$\int{\sin \left( 2\pi t \right)dt}=-\dfrac{1}{2\pi }\cos (u)+C$$ Now reverse the substitution which we used $$2\pi t=u$$ $$\int{\sin \left( 2\pi t \right)dt=}-\dfrac{1}{2\pi }\cos \left( 2\pi t \right)+C$$ Therefore, the integration of the given integral $$\sin \left( 2\pi t \right)dt$$ is $$-\dfrac{1}{2\pi }\cos \left( 2\pi t \right)+C$$ . **Note:** While solving the above question be careful with the integration part. Do remember the substitution method used here for future use. Try to solve the question step by step. To solve these types of questions, we should have the knowledge of trigonometric identities. Do not forget to reverse the substitution. After integration add the constant of integration in the result.