Life of \pi

Is $\pi = \frac{22}{7}$?

There was an interesting problem that had come up during one of the JEE exams.

It asked us to integrate,
$$I = \int_{0}^{1} \frac{x^4 (1-x)^4}{1+x^2} dx$$
Use Polynomial Long Division,
$$I=={\displaystyle\int}\left(-\dfrac{4}{x^2+1}+x^6-4x^5+5x^4-4x^2+4\right)\mathrm{d}x$$

Use Linearity,
$$=-\class{steps-node}{\cssId{steps-node-1}{4}}{\displaystyle\int}\dfrac{1}{x^2+1}\,\mathrm{d}x+$$

$${\displaystyle\int}x^6\,\mathrm{d}x-\class{steps-node}{\cssId{steps-node-2}{4}}{\displaystyle\int}x^5\,\mathrm{d}x+\class{steps-node}{\cssId{steps-node-3}{5}}{\displaystyle\int}x^4\,\mathrm{d}x-\class{steps-node}{\cssId{steps-node-4}{4}}{\displaystyle\int}x^2\,\mathrm{d}x+\class{steps-node}{\cssId{steps-node-5}{4}}{\displaystyle\int}1\,\mathrm{d}x$$

$$I=-4\arctan\left(x\right)+\dfrac{x^7}{7}-\dfrac{2x^6}{3}+x^5-\dfrac{4x^3}{3}+4x$$

Substitute the values and we will get,
$$I = -\dfrac{7{\pi}-22}{7} = -(\pi - \frac{22}{7})$$
Which comes to,
$$\displaystyle I = \frac{22}{7} - \pi \tag*{(1)}$$

Since, Area cannot be negative, it is evident from $(1)$ that,
$$\frac{22}{7} \geq \pi$$


Also, $\pi \neq \frac{22}{7}$ is evident from,
$$\boxed{ e^{\pi i}+1 \neq e^{\frac{22}{7} i}+1}$$
Cheers!