Question

How do you integrate \[\int{3dt}\]?

Answer 1

Hint: In the given question, we have been asked to integrate the given constant. In order to solve the question, we integrate the numerical by using the basic concept of integration. First we need to take out the constant part out of the integration. Later we will need to integrate the variable part using a suitable integration formula and we will get our required answer.

Complete step-by-step answer:
We have given,
\[\Rightarrow \int{3dt}\]
Let I be the integration of the given equation.
Therefore, we can write the integration as,
\[\Rightarrow I=\int{3dt}\]
Taking the constant part out of the integration, we get
\[\Rightarrow I=3\int{dt}\]
As we know that,
Integration of any constant ‘k’,
\[\Rightarrow \int{kf(t)dt}=k\int{f(t)dt}\]
Therefore,
\[\Rightarrow I=\int{3dt}=3\int{dt=3}\int{{{t}^{0}}dt}\]
Using the integration formula, i.e.
\[\Rightarrow \int{{{t}^{n}}dt=\dfrac{{{t}^{n+1}}}{n+1}}+c\], where n is not equal to -1,
Thus applying this formula, we have
\[\Rightarrow \int{3dt}=3\int{dt=3}\int{{{t}^{0}}dt}=3\left( \dfrac{{{t}^{0+1}}}{0+1} \right)=3\left( \dfrac{{{t}^{1}}}{1} \right)=3t\]
Therefore,
\[\Rightarrow I=\int{3dt}=3t+c\]
\[\therefore \int{3dt}=3t+c\]

Hence, the required integration is \[\int{3dt}=3t+c\]

Note: Here in this question, we need to use the linearity rules of integration. The linearity rule of the integration states that the integration of a constant will be a multiple of the given function. Therefore, the antiderivative or the integration of a constant \[a\]with respect to \[x\] is \[ax+C\].
The question given above its just a simple integration of a constant for that we should always remember to take out the constant term out of the integration and integrate the remaining integral by using the basic concepts or methods of integration. We should remember the property or the formulas of integration, this would make it easier to solve the question.