Question

What is the integral of $tan\left(\dfrac{\pi}{4}\right)ydy$?

Answer 1

We are given an integral for which we will use a certain set of formulae. Here, a constant is multiplied with y and the integral has been asked to find out. We simply use the polynomial integral formula along with the constant multiplication formula and solve it directly.

Complete step by step solution:
We are given $tan \left(\dfrac{\pi}{4}\right) \times ydy$
We see that $tan\dfrac{\pi}{4}$ is a constant. And while calculating integral, the constant has no effect on the integration of the term, so it can be taken out. In short, we can use the formula:
$\int(c\times a(x))dx=c\times\int a(x) dx$, where 'c' is a constant and a(x) is a function of 'x'.
So, till now we have separated the constant and the situation looks like the following:
$\int tan \left (\dfrac{\pi}{4}\right)y dy=tan\left(\dfrac{\pi}{4}\right)\int ydy$
Now after taking the constant out, we can do the integration left inside the integration. We use the following formula:
$\int xdx=\dfrac{x^2}{2}+c$, where 'c' is an arbitrary constant.
So, we have the following situation now:
$\int tan\left(\dfrac{\pi}{4}\right) ydy=tan\left(\dfrac{\pi}{4}\right)\times \dfrac{y^2}{2}+c$, where 'c' is an arbitrary constant.
Now, we put the value of $tan\dfrac{\pi}{4}$. We know that:
$tan\dfrac{\pi}{4}=1$
Using this we obtain the final result as:
$\int tan\left(\dfrac{\pi}{4}\right) ydy=\dfrac{y^2}{2}+c$
Hence, the integral is obtained.

Note: Only the constant can be taken out of the integral sign, any other term taken out would lead to an invalid answer. Moreover, the value of $tan\left(\dfrac{\pi}{4}\right)$ should be put correctly, do not get confused between the other tan values. Always take the constant out and then do the integration.