Question

Find $\int _{1}^{2}\left[ 3x \right]dx$ where $\left[ \centerdot \right]$ represents the greatest integer function, is?

Answer 1

To solve the given integral problem, we will be integrating by substitution method. Firstly, we will be finding the part that is going to be used and can be named as $u$ or $t$.Now this function is to be differentiated with respect to $x$ and then we are supposed to solve it. Now we will be rewriting all the integrals in terms of $u$ and after integrating, we will obtain our required answer.

Complete step-by-step answer:
Let us know why we are using the substitution method in our problem. This method is used when an integral contains the function and its derivative. In this case, we can set $u$ equal to the function and rewrite the integral in terms of the new variable $u$. And now we can find that, this makes it easier to solve.
Now let us find the greatest integer function of $\int _{1}^{2}\left[ 3x \right]dx$.
Firstly, make it equal to $I$ and rewrite the function.
We get, $I=\int _{1}^{2}\left[ 3x \right]dx$.
Now let us substitute $3x=t$
In the next step, we are supposed to differentiate it. On differentiating, we get-

& 3x=t \\\ & dx=\dfrac{1}{3}dt \\\ \end{aligned}$$ Now let us apply the rule of substitution method and solve it accordingly. $$I=\dfrac{1}{3}\int _{3}^{6}[t]dt$$ Let us integrate this by parts. $$\begin{aligned} & \Rightarrow \dfrac{1}{3}\left( \int _{3}^{4}\left[ t \right]dt+\int _{4}^{5}\left[ t \right]dt+\int _{5}^{6}\left[ t \right]dt \right) \\\ & =\dfrac{1}{3}\left( 3\int _{3}^{4}dt+4\int _{4}^{5}dt+5\int _{5}^{6}dt \right) \\\ \end{aligned}$$ Since we have done with integrating, let us apply the lower bound and upper bound values to the function. $$=\dfrac{1}{3}\left( 3\left[ t \right]_{3}^{4}+4\left[ t \right]_{4}^{5}+5\left[ t \right]_{5}^{6} \right)$$ Upon further solving this, we get $$\begin{aligned} & \dfrac{1}{3}\left( 3+4+5 \right) \\\ & =\dfrac{1}{3}\left( 12 \right) \\\ & =4 \\\ \end{aligned}$$ $$\therefore $$ The greatest integer function of $$\int _{1}^{2}\left[ 3x \right]dx$$ is $$4$$. **Note:** We have changed the limits while solving the integral as the substitution method requires it to be changed in the case of definite integrals. When we change variables in the integrand, limits of integration change as well.