Direct integration equations

#022

Problem

y' = \frac{1}{1 + x^2}

Classification

inverse trig pattern
solution family

Method

direct integration

Solution

y = \arctan x + C

Simulation & Code

Computational workSimulation

Download .py file

Python codeOpen

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-6, 6, 800)

with np.errstate(divide="ignore", invalid="ignore", over="ignore"):
    y = np.arctan(x)
    y_prime = 1/(1 + x**2)

plt.figure(figsize=(8, 5))
plt.plot(x, y, label="y = arctan(x)", linewidth=2)
plt.plot(x, y_prime, label="y' = 1/(1 + x^2)", linewidth=2, color="red")
plt.title("Entry #022: solution and derivative")
plt.xlabel("x")
plt.ylabel("value")
plt.grid(True, alpha=0.35)
plt.legend()
plt.tight_layout()
plt.show()

Handwritten derivation

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Takeaway & Interpretation

This equation says the solution's slope is controlled entirely by 1/(1 + x^2). After integrating, the solution family becomes y = arctan(x) + C, meaning every solution has the same overall shape but may be shifted vertically by the constant C.